U.S. patent number 10,230,724 [Application Number 15/353,633] was granted by the patent office on 2019-03-12 for identification method of an entity.
This patent grant is currently assigned to IDEMIA IDENTITY & SECURITY. The grantee listed for this patent is Safran Identity & Security. Invention is credited to Julien Bringer, Herve Chabanne, Roch Olivier Lescuyer De Chaptal-Lamure, Eduardo Soria-Vazquez.
![](/patent/grant/10230724/US10230724-20190312-D00000.png)
![](/patent/grant/10230724/US10230724-20190312-D00001.png)
![](/patent/grant/10230724/US10230724-20190312-D00002.png)
![](/patent/grant/10230724/US10230724-20190312-D00003.png)
![](/patent/grant/10230724/US10230724-20190312-D00004.png)
![](/patent/grant/10230724/US10230724-20190312-D00005.png)
![](/patent/grant/10230724/US10230724-20190312-D00006.png)
![](/patent/grant/10230724/US10230724-20190312-D00007.png)
![](/patent/grant/10230724/US10230724-20190312-D00008.png)
![](/patent/grant/10230724/US10230724-20190312-D00009.png)
![](/patent/grant/10230724/US10230724-20190312-D00010.png)
View All Diagrams
United States Patent |
10,230,724 |
Bringer , et al. |
March 12, 2019 |
Identification method of an entity
Abstract
A biometric identification method of an entity including
computation of a matching value between biometric data of an entity
u and reference biometric data u', by application of a function F
to the biometric data. A non-interactive, publicly verifiable
computation method is performed wherein representation of the
function is obtained by converting an arithmetic circuit into a
polynomial representation. A matching value is obtained by
evaluating the arithmetic circuit and the reference biometric data
as inputs. Proof of correction of the computation execution of the
matching values is obtained. Verification of said received proof.
The function is encoded with an integer k>1 of a vector of a
biometric datum on at least one input wire of the circuit. The
function includes at least m scalar products. Evaluation of the
circuit is iteratively computed depending on the value of m.
Inventors: |
Bringer; Julien (Issy les
Moulineaux, FR), Lescuyer De Chaptal-Lamure; Roch
Olivier (Issy les Moulineaux, FR), Chabanne;
Herve (Issy les Moulineaux, FR), Soria-Vazquez;
Eduardo (Issy les Moulineaux, FR) |
Applicant: |
Name |
City |
State |
Country |
Type |
Safran Identity & Security |
Issy-les-Moulineaux |
N/A |
FR |
|
|
Assignee: |
IDEMIA IDENTITY & SECURITY
(Issy les Moulineaux, FR)
|
Family
ID: |
55646697 |
Appl.
No.: |
15/353,633 |
Filed: |
November 16, 2016 |
Prior Publication Data
|
|
|
|
Document
Identifier |
Publication Date |
|
US 20170142103 A1 |
May 18, 2017 |
|
Foreign Application Priority Data
|
|
|
|
|
Nov 16, 2015 [FR] |
|
|
15 60942 |
|
Current U.S.
Class: |
1/1 |
Current CPC
Class: |
H04L
9/3231 (20130101); G06F 17/10 (20130101); H04L
9/3218 (20130101); H04L 63/0861 (20130101); H04L
63/0442 (20130101); G06F 21/32 (20130101); H04L
63/061 (20130101); H04L 9/0869 (20130101); H04L
2209/76 (20130101); H04L 2209/46 (20130101) |
Current International
Class: |
H04L
29/06 (20060101); G06F 17/10 (20060101); G06F
21/32 (20130101); H04L 9/32 (20060101); H04L
9/08 (20060101) |
Field of
Search: |
;713/171 |
References Cited
[Referenced By]
U.S. Patent Documents
Other References
Ben-Sasson et al. "Secure Sampling of Public Parameters for
Succinct Zero Knowledge Proofs" Sep. 25, 2015; 2015 IEEE Symposium
on Security and Privacy, pp. 287-304 (Year: 2015). cited by
examiner .
French Search Report with English Language Translation Cover Sheet
dated Jul. 27, 2016, FR Application No. 1560942. cited by applicant
.
Anonymous, "Efficient Way to Multiply a Large Set of Small
Numbers", Stack Overflow Site, Apr. 2014, pp. 5-8. cited by
applicant .
Barni, Mauro, et al., "Privacy Protection in Biometric-Based
Recognition Systems: A Marriage between Cryptography and Signal
Processing", IEEE Signal Processing Magazine, vol. 32, No. 5, Sep.
2015, pp. 70-76. cited by applicant .
Bringer, Julien, et al., "Gshade: Faster Privacy-Preserving
Distance Computation and Biometric Identification", Proceedings of
the 2nd ACM Workshop on Information Hiding and Multimedia Security,
IH&MMSEC '14, Jun. 2014, pp. 191-198. cited by applicant .
Bringer, Julien, et al., "Privacy-Preserving Biometric
Identification Using Secure Multiparty Computation: An Overview and
Recent Trends", IEEE Signal Processing Magazine, IEEE Service
Center, Piscataway, NJ, US, vol. 30, No. 2, Mar. 2013, pp. 46-52.
cited by applicant .
Bringer, Julien, et al., "Some Applications of Verifiable
Computation to Biometric Verification", 2015 IEEE International
Workshop on Information Forensics and Security (WIFS), IEEE, Nov.
2015, pp. 5-6. cited by applicant .
Costello, Craig, et al., "Geppetto: Versatile Verifiable
Computation", International Association for Cryptologic Research,
vol. 20141201:093827, Nov. 2014, pp. 5-21. cited by applicant .
Parno, Bryan, et al., "Pinnocchio: Nearly Practical Verifiable
Computation", Security and Privacy (SP), 2013 IEEE Symposium on,
IEEE, May 2013, pp. 238-252. cited by applicant .
Upmanyu, Maneesh, et al., "Efficient Biometric Verification in
Encrypted Domain", Advances in Biometrics, Springer Berlin
Heidelberg, Berlin, Heidelberg, Jun. 2009, pp. 903-908. cited by
applicant .
Walfish, Michael, et al., "Verifying Computations without
Reexecuting Them", Communications of the ACM, Association for
Computing Machinery, Inc, United States, vol. 58, No. 2, Jan. 2015,
pp. 78-84. cited by applicant.
|
Primary Examiner: Naghdali; Khalil
Attorney, Agent or Firm: Womble Bond Dickinson (US) LLP
Claims
The invention claimed is:
1. A biometric identification method of an entity, by a biometric
identification system comprising a client device and a remote
computation device, comprising: computation of at least one
matching value between at least one biometric datum of the entity u
and at least one reference biometric datum u', by application of a
function F to said biometric data, each of said data being a vector
of N binary integers u.sub.i or u'.sub.i with 1.ltoreq.i.ltoreq.N,
each integer being coded on n bits, said function F comprising a
scalar product between a biometric datum of the entity and a
reference biometric datum, said computation performing a
non-interactive, publicly verifiable computation method comprising
steps of: representation of said function F in form of an
arithmetic circuit comprising wires transporting values of finite
prime field .sub.q, with q a prime number, and connecting addition
and multiplication operators, conversion of said arithmetic circuit
into a polynomial representation, QAP (Quadratic Arithmetic
Program) or multi-QAP, generation of a public evaluation key and of
a public verification key as a function of said polynomial
representation, obtaining by the remote computation device of the
arithmetic circuit and of the public evaluation key, for each
biometric datum of the entity, determination of at least one
matching value between said biometric datum and at least one
reference biometric datum by the remote computation device by
evaluating the arithmetic circuit having as inputs the biometric
datum of the entity and the reference biometric datum, for each
determined matching value, generation by the remote computation
device of a proof of correction of computation execution of
matching value, so-called generated proof, from said polynomial
representation, the public evaluation key and result of the
evaluation of the arithmetic circuit, transmission by the remote
computation device of matching values and generated proofs to the
client device, verification of said generated proofs received by
the client device by means of the public verification key,
identification of the entity by the client device as the function
of the matching values and result of said verification of the
generated proofs, wherein: representation of said function F
comprises encoding an integer k>1 of binary integers of a vector
of a biometric datum on at least one input wire of the arithmetic
circuit, and the function F comprising at least m scalar products,
m being a divider of length N of biometric data vectors, if the
divider m is equal to 2 or 3, the arithmetic circuit comprises at
least N/(k*m) multiplication operators connected to input wires of
the arithmetic circuit, a storage memory, and at least one addition
operator, and evaluation of the arithmetic circuit iteratively
comprises computation of each of the m scalar products by means of
said N/(k*m) multiplication operators, storage of m results of
computations of said scalar products in said storage memory and
summation of said results by means of said addition operator, if
the divider m is greater than or equal to 4, the arithmetic circuit
comprises at least one first computation sub-circuit of the scalar
product comprising N/(k*m) first multiplication operators connected
to input wires of the arithmetic circuit and a first storage
memory, and a second computation sub-circuit of the scalar product
comprising N/(k*m) second multiplication operators connected to the
input wires of the arithmetic circuit and a second storage memory,
each one of said first and second computation sub-circuits being
also connected to an output of the storage memory of the other one
of said first and second computation sub-circuits, and evaluation
of the arithmetic circuit iteratively comprises computation of each
of the m scalar products by using alternatively one of said first
and second computation sub-circuit to compute sum of the scalar
product of values of the input wires of the used one of said first
and second computation sub-circuits and value stored in the storage
memory of the other one of said first and second computation
sub-circuits, wherein said biometric identification method
delegates to a remote entity a comparison of biometric datum in
terms of a protocol of verifiable computations suitable for a
real-time execution.
2. The identification method according to claim 1, wherein the
verification of said received proofs comprises batch verification
of pairings.
3. The identification method according to claim 1, wherein: if the
divider m of the length N of the biometric data vectors is equal to
1, given an asymmetric bilinear environment (q, G.sub.1, G.sub.2,
G.sub.T, g.sub.1, g.sub.2, e) where q is a prime number G.sub.1,
G.sub.2 and G.sub.T three groups of order q, g.sub.1 a generator of
G.sub.1, g.sub.2 a generator of G.sub.2, and e a non-degenerate
bilinear pairing e: G.sub.1.times.G.sub.2.fwdarw.G.sub.T and the
arithmetic circuit being represented in form of a QAP of the
circuit Q=(t, V, W, Y) of size .rho. and degree .delta., with
V={vi}, W={wi}, Y={yi}, 0.ltoreq.i.ltoreq..rho., and given
I.sub.io={1, . . . , .theta.} set of indices corresponding to
input/output wires of the arithmetic circuit and
I.sub.mid={.theta.+1, . . . , .rho.} set of indices of intermediate
wires of the arithmetic circuit not being input wires of the
arithmetic circuit, the generation of the public evaluation key and
of the public verification key comprises: generation of random
variables r.sub.v, r.sub.w, s, .alpha..sub.v, .alpha..sub.w,
.alpha..sub.y, .beta., .gamma. in .sub.q, definition of
coefficients r.sub.y=r.sub.vr.sub.w, g.sub.v1=g.sub.1.sup.r.sup.v,
g.sub.w1=g.sub.1.sup.r.sup.w, g.sub.w2=g.sub.2.sup.r.sup.w,
g.sub.y1=g.sub.1.sup.r.sup.y and g.sub.y2=g.sub.2.sup.r.sup.y,
generation of public evaluation key EK.sub.F equal to (EK.sub.F1,
EK.sub.F2) where .times..times..times..times..function..di-elect
cons..times..times..function..di-elect
cons..times..times..function..di-elect
cons..times..times..alpha..function..di-elect
cons..times..times..alpha..function..di-elect
cons..times..times..alpha..function..di-elect cons..di-elect
cons..delta..times..times..beta..function..times..times..beta..function..-
times..times..beta..function..di-elect cons. ##EQU00035##
.times..times..times..times..function..di-elect cons.
##EQU00035.2## generation of public verification key VK.sub.F equal
to (VK.sub.F1, VK.sub.F2) where:
VK.sub.F1=(g.sub.1,{g.sub.v1.sup.v.sup.i.sup.(s)}.sub.i.di-elect
cons.[1,.theta.],{g.sub.y1.sup.y.sup.i.sup.(s)}.sub.i.di-elect
cons.[1,.theta.])
VK.sub.F2=(g.sub.2,g.sub.2.sup..alpha..sup.v,g.sub.2.sup..alpha..sup.w,g.-
sub.2.sup..alpha..sup.y,g.sub.2.sup..gamma.,g.sub.2.sup..beta..gamma.,g.su-
b.y2.sup.t(s),{g.sub.w2.sup.w.sup.i.sup.(s)}.sub.i.di-elect
cons.[1,.theta.]) generation by the remote computation device of
the proof of correction of the computation execution of a matching
value comprises, {c.sub.i}.sub.i.di-elect cons.[1,.rho.] being set
of values of the arithmetic circuit determined during determination
of the determined matching value: determination of a polynomial
h(x) such that p(x)=h(x)t(x) with
p(x)=(v.sub.0(x)+.SIGMA..sub.i=1.sup..rho.c.sub.iv.sub.i
(x))(w.sub.0(x)+.SIGMA..sub.i=1.sup..rho.c.sub.iw.sub.i(x))-(y.sub.0(x)+.-
SIGMA..sub.i=1.sup..rho.c.sub.iy.sub.i(x)), computation of
generated proof .pi.=(.pi..sub.1, .pi..sub.2) with:
.pi..times..times..function..times..times..function..times..times..functi-
on..function..times..times..alpha..function..times..times..alpha..function-
..times..times..alpha..function..times..times..beta..function..times..time-
s..beta..function..times..times..beta..function..times..pi..times..times..-
function. ##EQU00036## where: v.sub.mid(x)=.SIGMA..sub.i.di-elect
cons.I.sub.midc.sub.iv.sub.i(x),w.sub.mid(x)=.SIGMA..sub.i.di-elect
cons.I.sub.midc.sub.iw.sub.i(x) and
y.sub.mid(x)=.SIGMA..sub.i.di-elect
cons.I.sub.midc.sub.iy.sub.i(x), and, wherein said proofs received
by the client device being equal to (.pi..sub.r1, .pi..sub.r2) with
.pi..sub.r1 in form of: (g.sub.v1.sup.V.sup.mid,
g.sub.w1.sup.W.sup.mid, g.sub.y1.sup.Y.sup.mid, g.sub.1.sup.H,
g.sub.v1.sup.V'.sup.mid, g.sub.w1.sup.W'.sup.mid,
g.sub.y1.sup.Y'.sup.mid, g.sub.1.sup.Z) and .pi..sub.r2 in the form
g.sub.w2.sup.w.sup.mid, the verification of a received proof
comprises performing following equality tests:
e(g.sub.v1.sup.v.sup.o.sup.(s)g.sub.v1.sup.v.sup.io.sup.(s)g.sub.v1.sup.V-
.sup.mid,g.sub.w2.sup.w.sup.o.sup.(s)g.sub.w2.sup.w.sup.io.sup.(s)g.sub.w2-
.sup.W.sup.mid)=e(g.sub.1.sup.H,g.sub.y2.sup.t(s))e(g.sub.y1.sup.y.sup.o.s-
up.(s)g.sub.y1.sup.y.sup.io.sup.(s)g.sub.y1.sup.Y.sup.mid,g.sub.2),
e((g.sub.v1.sup.V'.sup.mid).sup.d.sup.1(g.sub.w1.sup.W'.sup.mid).sup.d.su-
p.2(g.sub.y1.sup.Y'.sup.mid).sup.d.sup.3,g.sub.2)=e((g.sub.v1.sup.V.sup.mi-
d).sup.d.sup.1,g.sub.2.sup..alpha..sup.v)e((g.sub.w1.sup.W.sup.mid).sup.d.-
sup.2,g.sub.2.sup..alpha..sup.w)e((g.sub.y1.sup.Y.sup.mid).sup.d.sup.3,g.s-
ub.2.sup..alpha..sup.y),e((g.sub.1.sup.Z,g.sub.2.sup..gamma.)=e(g.sub.v1.s-
up.V.sup.midg.sub.w1.sup.W.sup.midg.sub.y1.sup.Y.sup.mid,g.sub.2.sup..beta-
..gamma.) where
g.sub.v1.sup.v.sup.io.sup.(s)=.PI..sub.i=1.sup..theta.(g.sub.v1.sup.v.sup-
.i.sup.(s)).sup.c.sup.i,
g.sub.w1.sup.w.sup.io.sup.(s)=.PI..sub.i=1.sup..theta.(g.sub.w1.sup.w.sup-
.i.sup.(s)).sup.c.sup.i, and
g.sub.y1.sup.y.sup.io.sup.(s)=.PI..sub.i=1.sup..theta.(g.sub.y1.sup.y.sup-
.i.sup.(s)).sup.c.sup.i and (d.sub.1, d.sub.2, d.sub.3) elements of
.sub.q on .lamda. bits with .lamda. a security parameter.
4. The identification method according to claim 1 wherein: if the
divider m of the length N of the biometric data vectors is greater
than or equal to 2, given an asymmetric bilinear environment (q,
G.sub.1, G.sub.2, G.sub.T, g.sub.1, g.sub.2, e) where q is a prime
number G.sub.1, G.sub.2 and G.sub.T three groups of order q,
g.sub.1 a generator of G.sub.1, g.sub.2 a generator of G.sub.2, and
e a non-degenerate bilinear pairing e:
G.sub.1.times.G.sub.2.fwdarw.G.sub.T, the arithmetic circuit being
represented in form of a multi-QAP Q=({B.sub.b}.sub.b.di-elect
cons.[1,l],t,V,W, of size .rho. and degree .delta., with
{B.sub.b}.sub.b.di-elect cons.[1,l] a set of l banks B.sub.b of Q
used in the computation of the function F, and V={vi}, W={wi},
Y={yi} with 0.ltoreq.i.ltoreq..rho., the generation of the public
evaluation key and of the public verification key comprises:
generation of random variables s, {(.alpha..sub.bv, .alpha..sub.bw,
.alpha..sub.by, .beta..sub.b,.gamma..sub.b)}.sub.b.di-elect
cons.[1,l], r.sub.v, r.sub.w in .sub.q, definition of following
coefficients: r.sub.y=r.sub.yr.sub.w, g.sub.v1=g.sub.1.sup.r.sup.v,
g.sub.v2=g.sub.2.sup.r.sup.v, g.sub.w1=g.sub.1.sup.r.sup.w,
g.sub.w2=g.sub.2.sup.r.sup.w, g.sub.y1=g.sub.1.sup.r.sup.y and
g.sub.y2=g.sub.2.sup.r.sup.y, generation of public evaluation key
EK.sub.F equal to: ({EK.sub.Fb}.sub.b.di-elect
cons.[1,l],{g.sub.1.sup.s.sup.i}.sub.i.di-elect
cons.[1,.delta.],g.sub.v1.sup.t(s),g.sub.w1.sup.t(s),g.sub.y1.sup.t(s),g.-
sub.v2.sup.t(s),g.sub.w2.sup.t(s),g.sub.y2.sup.t(s)) where each
public bank key EK.sub.Fb is equal to (EK.sub.Fb1, EK.sub.Fb2)
with:
.times..times..times..times..function..times..times..function..times..tim-
es..function..times..times..alpha..function..times..times..alpha..function-
..times..times..alpha..function..times..times..beta..function..times..time-
s..beta..function..times..times..beta..function..di-elect
cons..times..times..alpha..function..times..times..alpha..function..times-
..times..alpha..function..times..times..beta..function..times..times..beta-
..function..times..times..beta..function. ##EQU00037##
EK.sub.Fb2=({g.sub.w2.sup.w.sup.i.sup.(s)}.sub.i.di-elect
cons.B.sub.b,g.sub.w2.sup..alpha..sup.bw.sup.,t(s)) generation of
the public verification key VK.sub.F equal to:
({VK.sub.Fb}.sub.b.di-elect cons.[1,l], g.sub.1, g.sub.2,
g.sub.y2.sup.t(s)) where each public bank key VK.sub.Fb is equal to
(g.sub.2.sup..alpha..sup.bv, g.sub.2.sup..alpha..sup.bw,
g.sub.2.sup..alpha..sup.by, g.sub.2.sup..gamma..sup.b,
g.sub.2.sup..beta..sup.b.sup..gamma..sup.b), determination of a
matching value comprises, function F being divided into .omega.
sub-functions F.sub.1, . . . , F.sub..omega. and .sigma.=((f.sub.l,
(T.sub.l1, . . . , T.sub.ll))).sub.l.di-elect cons.[1,L] being a
scheduling of length L with f.sub.l .di-elect cons.{1, . . . ,
.omega.}, evaluation of each sub-function F.sub..omega. from the
biometric data of the entity and the reference biometric data and
determination of the values of the arithmetic circuit, generation
by the remote computation device of the proof of correction of the
computation execution of a matching value comprises, for each l={1,
. . . , L}: for each bank B.sub.b such that b.di-elect
cons..LAMBDA., with .LAMBDA.[1,l] set of indices b.di-elect
cons.[1,l] such that T.sub.lb.noteq.0,.GAMMA.=U.sub.b.di-elect
cons..LAMBDA.B.sub.b, {c.sub.j}.sub.j.di-elect cons.B.sub.b an
instance of bank B.sub.b, .DELTA.={c.sub.i}.sub.i.di-elect
cons..GAMMA. set of values of .GAMMA.: generation of pledging
random variables in .sub.q: o.sub.b=(o.sub.bv, o.sub.bw, o.sub.by),
computation of a digest D.sub.b equal to (D.sub.b1,D.sub.b2) from
the instance of the bank B.sub.b:
B.sub.b.sup.(T.sup.lb.sup.)={c.sub.i.di-elect
cons..DELTA.}.sub.i.di-elect cons.B.sub.b and pledging random
variables o.sub.b and such that: if the bank B.sub.b is an
input/output bank:
D.sub.b1=(g.sub.v1.sup.v.sup.(b).sup.(s),g.sub.y1.sup.y.sup.(b).sup.(s))
and D.sub.b2=(g.sub.w2.sup.w.sup.(b).sup.(s)), if the bank B.sub.b
is not input/output bank:
.times..times..times..times..function..times..times..function..times..tim-
es..function..times..times..alpha..function..times..times..alpha..function-
..times..times..alpha..function..times..times..beta..function..times..time-
s..beta..function..times..times..beta..function. ##EQU00038##
.times..times..times..times..function. ##EQU00038.2## with:
v.sup.(b)(s)=.SIGMA..sub.i.di-elect
cons.B.sub.bc.sub.iv.sub.i(s)+o.sub.bvt(s),
w.sup.(b)(s)=.SIGMA..sub.i.di-elect
cons.B.sub.bc.sub.iw.sub.i(s)+o.sub.bwt(s),
y.sup.(b)(s)=.SIGMA..sub.i.di-elect
cons.B.sub.bc.sub.iy.sub.i(s)+o.sub.byt(s), determination of a
polynomial h.sup.(l)(x) such that p.sup.(l)(x)=h.sup.(l)(x)t(x)
with p.sup.(l)(x)=(v.sub.0(x)+.SIGMA..sub.i.di-elect
cons..GAMMA.c.sub.iv.sub.j(x)+.SIGMA..sub.b.di-elect
cons..LAMBDA.o.sub.bvt(x))(w.sub.0(x)+.SIGMA..sub.i.di-elect
cons..GAMMA.c.sub.iw.sub.j(x)+.SIGMA..sub.b.di-elect
cons..LAMBDA.o.sub.bwt(x))-(y.sub.0(x)+.SIGMA..sub.i.di-elect
cons..GAMMA.c.sub.iy.sub.j(x)+.SIGMA..sub.b.di-elect
cons..LAMBDA.o.sub.byt(x)) computation of a proof element
.pi..sup.(l) equal to g.sub.1.sup.h.sup.(l).sup.(s), and, wherein
said proofs received by the client device being of the form
D.sub.1.sup.(1), . . . , D.sub.l.sup.(1), .pi..sup.(1), . . . ,
D.sub.1.sup.(L), . . . , D.sub.l.sup.(L), .pi..sup.(L) where for
all l.di-elect cons.{1, . . . , L} and b.di-elect cons.{1, . . .
,l}:
.times..times..times..times..times..times..times..times.'.times..times..t-
imes..times.'.times..times..times..times.'.times..times..times..times.
##EQU00039## and .pi..sup.(l)=g.sub.1.sup.H.sup.(l), verification
of the received proof comprises: verification of Ll digests, for
l.di-elect cons.{1, . . . , L} and b.di-elect cons.{1, . . . , l}
comprising performing following equality tests:
.function..times..times.'.times..times..alpha..function..times..times.
##EQU00040##
.function..times..times.'.function..alpha..function..times..times.
##EQU00040.2##
.function..times..times.'.times..times..alpha..function..times..times.
##EQU00040.3##
.function..gamma..function..times..times..times..times..times..times..bet-
a..gamma. ##EQU00040.4## verification of L proofs comprising for
l.di-elect cons.{1, . . . , L} performing following equality test:
.function. .times..times..times.
.times..times..times..times..function..times..times..function..function.
.times..times..times. ##EQU00041##
5. The identification method according to claim 1 wherein: if the
divider m of the length N of the biometric data vectors is greater
than or equal to 2, given an asymmetric bilinear environment (q,
G.sub.1, G.sub.2, G.sub.T, g.sub.1, g.sub.2, e) where q is a prime
number G.sub.1, G.sub.2 and G.sub.T three groups of order q,
g.sub.1 a generator of G.sub.1, g.sub.2 a generator of G.sub.2, and
e a non-degenerate bilinear pairing e:
G.sub.1.times.G.sub.2.fwdarw.G.sub.T, the arithmetic circuit being
represented in form of a multi-QAP Q=({B.sub.b}.sub.b.di-elect
cons.[1,l], t, V, W, of size .rho. and degree .delta., with
{B.sub.b}.sub.b.di-elect cons.[1,l] a set of l banks B.sub.b of Q
used in computation of the function F, and V={vi}, W={wi}, Y={yi}
with 0.ltoreq.i.ltoreq..rho., the generation of the public
evaluation key and the public verification key comprises:
generation of random variables
s,{(.alpha..sub.bv,.alpha..sub.bw,.alpha..sub.by, .beta..sub.b,
.gamma..sub.b)}.sub.b.di-elect cons.[1,l], r.sub.v, r.sub.w in
.sub.q, generation of random variables s,{(.alpha..sub.bv,
.alpha..sub.bw, .alpha..sub.by, .beta..sub.b,
.gamma..sub.b)}.sub.b.di-elect cons.[1,l], r.sub.v, r.sub.w in
.sub.q, definition of the following coefficients:
r.sub.y=r.sub.br.sub.w, g.sub.v1=g.sub.1.sup.r.sup.v,
g.sub.v2=g.sub.2.sup.r.sup.v, g.sub.w1=g.sub.1.sup.r.sup.w,
g.sub.w2.sup.r.sup.w=g.sub.2.sup.r.sup.w,
g.sub.y1=g.sub.1.sup.r.sup.y and g.sub.y2=g.sub.2.sup.r.sup.y,
generation of the public evaluation key EK.sub.F equal to:
({EK.sub.Fb}.sub.b.di-elect
cons.[1,l],{g.sub.1.sup.s.sup.i}.sub.i.di-elect
cons.[1,.delta.],g.sub.v1.sup.t(s),g.sub.w1.sup.t(s),g.sub.y1.sup.t(s),g.-
sub.v2.sup.t(s),g.sub.w2.sup.t(s),g.sub.y2.sup.t(s)) where each
public bank key EK.sub.Fb is equal to (EK.sub.Fb1, EK.sub.Fb2)
with:
.times..times..times..times..function..times..times..function..times..tim-
es..function..times..times..alpha..function..times..times..alpha..function-
..times..times..alpha..function..times..times..beta..function..times..time-
s..beta..function..times..times..beta..function..di-elect
cons..times..times..alpha..function..times..times..alpha..function..times-
..times..alpha..function..times..times..beta..function..times..times..beta-
..function..times..times..beta..function. ##EQU00042##
.times..times..times..times..times..function..di-elect
cons..times..times..alpha..function. ##EQU00042.2## generation of
the public verification key VK.sub.F is equal to:
({VK.sub.Fb}.sub.b.di-elect cons.[1,l], g.sub.1, g.sub.2,
g.sub.y2.sup.t(s)) where each public bank key VK.sub.Fb is equal to
(g.sub.2.sup..alpha..sup.bv, g.sub.2.sup..alpha..sup.bw,
g.sub.2.sup..alpha..sup.by, g.sub.2.sup..gamma..sup.b,
g.sub.2.sup..beta..sup.b.sup..gamma..sup.b), determination of the
matching value comprises, function F being divided into .omega.
sub-functions F.sub.1, . . . , F.sub..omega. and .sigma.=((f.sub.l,
(T.sub.l1, . . . , T.sub.ll))).sub.l.di-elect cons.[1,L] being a
scheduling of length L with f.sub.l .di-elect cons.{1, . . . ,
.omega.}, evaluation of each sub-function F.sub..omega. from the
biometric data of the entity and the reference biometric data and
determination of the determined values of the arithmetic circuit,
generation by the remote computation device of the proof of
correction of the computation execution of the matching value
comprises, for each l={1, . . . , L}: for each bank B.sub.b such
that b.di-elect cons..LAMBDA., with .LAMBDA.[1,l] the set of
indices b.di-elect cons.[1,l] such that
T.sub.lb.noteq.0,.GAMMA.=U.sub.b.di-elect cons..LAMBDA.B.sub.b,
{c.sub.j}.sub.j.di-elect cons.B.sub.b an instance of the bank
B.sub.b, .DELTA.={c.sub.i}.sub.i.di-elect cons..GAMMA. the set of
values of .GAMMA.: generation of pledging random variables in
.sub.q: o.sub.b=(o.sub.bv, o.sub.bw, o.sub.by), computation of a
digest D.sub.b equal to (D.sub.b1,D.sub.b2) from the instance of
the bank B.sub.b: B.sub.b.sup.(T.sup.lb.sup.)={c.sub.i.di-elect
cons..DELTA.}.sub.i.di-elect cons.B.sub.b and pledging random
variables o.sub.b and such that: if the bank B.sub.b is
input/output bank:
D.sub.b1=(g.sub.v1.sup.v.sup.(b).sup.(s),g.sub.y1.sup.y.sup.(b).sup.(s))
and D.sub.b2=(g.sub.w2.sup.w.sup.(b).sup.(s)), if the bank B.sub.b
is not the input/output bank:
.times..times..times..times..function..times..times..function..times..tim-
es..function..times..times..alpha..function..times..times..alpha..function-
..times..times..alpha..function..times..times..beta..function..times..time-
s..beta..function..times..times..beta..function. ##EQU00043##
.times..times..times..times..function. ##EQU00043.2## with:
v.sup.(b)(s)=.SIGMA..sub.i.di-elect
cons.B.sub.bc.sub.iv.sub.i(s)+o.sub.bvt(s),
w.sup.(b)(s)=.SIGMA..sub.i.di-elect
cons.B.sub.bc.sub.iw.sub.i(s)+o.sub.bwt(s),
y.sup.(b)(s)=.SIGMA..sub.i.di-elect
cons.B.sub.bc.sub.iy.sub.i(s)+o.sub.byt(s), determination of a
polynomial h.sup.(l)(x) such that p.sup.(l)(x)=h.sup.(l)(x)t(x)
with p.sup.(l)(x)=(v.sub.0(x)+.SIGMA..sub.i.di-elect
cons..GAMMA.c.sub.iv.sub.j(x)+.SIGMA..sub.b.di-elect
cons..LAMBDA.o.sub.bvt(x))(w.sub.0(x)+.SIGMA..sub.i.di-elect
cons..GAMMA.c.sub.iw.sub.j(x)+.SIGMA..sub.b.di-elect
cons..LAMBDA.o.sub.bwt(x))-(y.sub.0(x)+.SIGMA..sub.i.di-elect
cons..GAMMA.c.sub.iy.sub.j(x)+.SIGMA..sub.b.di-elect
cons..LAMBDA.o.sub.byt(x)) computation of the proof element
.pi..sup.(l) equal to g.sub.1.sup.h.sup.(l).sup.(s), and, wherein
said proofs received by the client device being of form
D.sub.1.sup.(1), . . . , D.sub.l.sup.(1), .pi..sup.(1), . . . ,
D.sub.1.sup.(L), . . . , D.sub.l.sup.(L), .pi..sup.(L) where for
all l.di-elect cons.{1, . . . , L} and b.di-elect cons.{1, . . .
,l}:
.times..times..times..times..times..times..times..times.'.times..times..t-
imes..times.'.times..times..times..times.'.times..times..times..times.
##EQU00044## and .pi..sup.(l)=g.sub.1.sup.H.sup.(l), verification
of the received proof comprises, given a correction parameter
.lamda.: selection of a random vector (d.sub.1, . . . , d.sub.3l)
of elements of size .lamda., batch verification of Ll digests
comprising performing l times the following equality tests, for
b.di-elect cons.{1, . . . , l}:
.function..times..times..times.'.times..times..alpha..function..times..ti-
mes..times.'.times..times..alpha..function..times..times..times.'.times..t-
imes..alpha..function..times..times..times..times..times..times..times..ti-
mes..times. ##EQU00045##
.times..function..times..gamma..function..times..times..times..times..tim-
es..times..times..beta..gamma. ##EQU00045.2## batch verification of
the L proofs comprising performing the following equality test:
.times..function. .times..times..times..times.
.times..times..times..function..times..times..times..function..function..-
times. .times..times..times. ##EQU00046##
6. The identification method according to claim 1, wherein the
identification of the entity comprises comparison of the matching
values with a predetermined threshold.
7. The identification method according to claim 1, wherein the
function F comprises comparison of result of the scalar product
between said biometric data of the entity and said reference
biometric data with a predetermined threshold.
8. The identification method according to claim 1, wherein the
encoding of k binary integers u.sub.i or u'.sub.i on an input wire
of an j.sup.th multiplication operator, 1.ltoreq.j.ltoreq.N/k, is
equal to
.times..times..function..times..times..times..times..times..times..functi-
on.' ##EQU00047## with 1.ltoreq.z.ltoreq.m, and is given by the
formula: .times..function..times..times..times..times. ##EQU00048##
with .epsilon..sub.1, . . . , .epsilon..sub.k predetermined
integers.
9. A computer program product comprising code instructions for
execution of a method according to a biometric identification
method of an entity, by a biometric identification system
comprising a client device and a remote computation device,
comprising: computation of at least one matching value between at
least one biometric datum of the entity u and at least one
reference biometric datum u', by application of a function F to
said biometric data, each of said data being a vector of N binary
integers u.sub.i or u'.sub.i with 1.ltoreq.i.ltoreq.N, each integer
being coded on n bits, said function F comprising a scalar product
between a biometric datum of the entity and a reference biometric
datum, said computation performing a non-interactive, publicly
verifiable computation method comprising steps of: representation
of said function F in form of an arithmetic circuit comprising
wires transporting values of finite prime field .sub.q, with q a
prime number, and connecting addition and multiplication operators,
conversion of said arithmetic circuit into a polynomial
representation, QAP (Quadratic Arithmetic Program) or multi-QAP,
generation of a public evaluation key and of a public verification
key as a function of said polynomial representation, obtaining by
the remote computation device of the arithmetic circuit and of the
public evaluation key, for each biometric datum of the entity,
determination of at least one matching value between said biometric
datum and at least one reference biometric datum by the remote
computation device by evaluating the arithmetic circuit having as
inputs the biometric datum of the entity and the reference
biometric datum, for each determined matching value, generation by
the remote computation device of a proof of correction of
computation execution of matching value, so-called generated proof,
from said polynomial representation, the public evaluation key and
result of the evaluation of the arithmetic circuit, transmission by
the remote computation device of matching values and generated
proofs to the client device, verification of said generated proofs
received by the client device by means of the public verification
key, identification of the entity by the client device as the
function of the matching values and result of said verification of
the generated proofs, wherein: representation of said function F
comprises encoding an integer k>1 of binary integers of a vector
of a biometric datum on at least one input wire of the arithmetic
circuit, and the function F comprising at least m scalar products,
m being a divider of length N of biometric data vectors, if the
divider m is equal to 2 or 3, the arithmetic circuit comprises at
least N/(k*m) multiplication operators connected to the input wires
of the arithmetic circuit, a storage memory, and at least one
addition operator, and evaluation of the arithmetic circuit
iteratively comprises computation of each of the m scalar products
by means of said N/(k*m) multiplication operators, storage of m
results of computations of said scalar products in said storage
memory and summation of said results by means of said addition
operator, if the divider m is greater than or equal to 4, the
arithmetic circuit comprises at least one first computation
sub-circuit of the scalar product comprising N/(k*m) first
multiplication operators connected to input wires of the arithmetic
circuit and a first storage memory, and a second computation
sub-circuit of the scalar product comprising N/(k*m) second
multiplication operators connected to the input wires of the
arithmetic circuit and a second storage memory, each one of said
first and second computation sub-circuits being also connected to
an output of the storage memory of the other one of said first and
second computation sub-circuits, and evaluation of the arithmetic
circuit iteratively comprises computation of each of the m scalar
products by using alternatively one of said first and second
computation sub-circuit to compute sum of the scalar product of
values of the input wires of the used one of said first and second
computation sub-circuits and value stored in the storage memory of
the other one of said first and second computation sub-circuits,
wherein said biometric identification method delegates to a remote
entity a comparison of biometric datum in terms of a protocol of
verifiable computations suitable for a real-time execution, when
said computer program product is executed by a processor.
10. A biometric identification system comprising a client device
and a remote computation device wherein: said client device and
said remote computation device each comprise a processor, an
interface and a memory for performing steps of an identification
method according to a biometric identification method of an entity,
by a biometric identification system comprising a client device and
a remote computation device, comprising: computation of at least
one matching value between at least one biometric datum of the
entity u and at least one reference biometric datum u', by
application of a function F to said biometric data, each of said
data being a vector of N binary integers u.sub.i or u'.sub.i with
1.ltoreq.i.ltoreq.N, each integer being coded on n bits, said
function F comprising a scalar product between a biometric datum of
the entity and a reference biometric datum, said computation
performing a non-interactive, publicly verifiable computation
method comprising steps of: representation of said function F in
form of an arithmetic circuit comprising wires transporting values
of the finite prime field .sub.q, with q a prime number, and
connecting addition and multiplication operators, conversion of
said arithmetic circuit into a polynomial representation, QAP
(Quadratic Arithmetic Program) or multi-QAP, generation of a public
evaluation key and of a public verification key as a function of
said polynomial representation, obtaining by the remote computation
device of the arithmetic circuit and of the public evaluation key,
for each biometric datum of the entity, determination of at least
one matching value between said biometric datum and at least one
reference biometric datum by the remote computation device by
evaluating the arithmetic circuit having as inputs the biometric
datum of the entity and the reference biometric datum, for each
determined matching value, generation by the remote computation
device of a proof of correction of computation execution of
matching value, so-called generated proof, from said polynomial
representation, the public evaluation key and result of the
evaluation of the arithmetic circuit, transmission by the remote
computation device of matching values and generated proofs to the
client device, verification of said generated proofs received by
the client device by means of the public verification key,
identification of the entity by the client device as the function
of the matching values and result of said verification of the
generated proofs, wherein: representation of said function F
comprises encoding an integer k>1 of binary integers of a vector
of a biometric datum on at least one input wire of the arithmetic
circuit, and the function F comprising at least m scalar products,
m being a divider of length N of biometric data vectors, if the
divider m is equal to 2 or 3, the arithmetic circuit comprises at
least N/(k*m) multiplication operators connected to input wires of
the arithmetic circuit, a storage memory, and at least one addition
operator, and evaluation of the arithmetic circuit iteratively
comprises computation of each of the m scalar products by means of
said N/(k*m) multiplication operators, storage of m results of
computations of said scalar products in said storage memory and
summation of said results by means of said addition operator, if
the divider m is greater than or equal to 4, the arithmetic circuit
comprises at least one first computation sub-circuit of the scalar
product comprising N/(k*m) first multiplication operators connected
to input wires of the arithmetic circuit and a first storage
memory, and a second computation sub-circuit of the scalar product
comprising N/(k*m) second multiplication operators connected to the
input wires of the arithmetic circuit and a second storage memory,
each one of said first and second computation sub-circuits being
also connected to an output of the storage memory of the other one
of said first and second computation sub-circuits, and evaluation
of the arithmetic circuit iteratively comprises computation of each
of the m scalar products by using alternatively one of said first
and second computation sub-circuit to compute sum of the scalar
product of values of the input wires of the used one of said first
and second computation sub-circuits and value stored in the storage
memory of the other one of said first and second computation
sub-circuits, wherein said biometric identification method
delegates to a remote entity a comparison of biometric datum in
terms of a protocol of verifiable computations suitable for a
real-time execution.
Description
GENERAL TECHNICAL FIELD
The present invention relates to the field of identification of an
entity, individual or object. More precisely, it relates to a
biometric identification method in which comparison to reference
biometric data is delegated to a remote device in terms of a
publicly verifiable computation protocol.
STATE OF THE ART
Due to increasing miniaturization of digital computation systems,
from now on there is a wide range of digital devices fitted with
variable computational power, including the smart card, the
supercomputer, the smartphone and the personal computer. In such a
scope it can be interesting for a device fitted with limited
computational power to delegate costly computations to a remote
device fitted with greater computational power. Such delegation in
particular has been made easier recently by the development of
high-speed communications networks and an offer to outsourced
computations to the cloud.
But such delegation of computations raises the problem of the
confidence which can be accorded to computation results transmitted
by a remote executing party. Computation errors can in fact occur,
also due to technical problems independent of the will of the
executing party, due to the fact of voluntary and malicious
action.
In this way, considerable efforts have been made to develop a
computation protocol, called verifiable computation, letting a
remote executing party prove to the device having ordered
computation that the latter was executed correctly. For a long time
however, developed protocols have remained applicable to specific
functions only, or else remained unusable in practice due to the
substantial number of computations necessary for the ordering party
to verify proof supplied by the executing party.
The Pinocchio protocol presented in the publication "Bryan Parno,
Craig Gentry, Jon Howell, and Mariana Raykova, Pinocchio: Nearly
Practical Verifiable Computation, in Proceedings of the IEEE
Symposium on Security and Privacy, IEEE, 21 May 2013" was one of
the first verifiable computation protocols for the executing party
to verifiably compute the application of any function and for the
ordering party to verify the associated proof in a computation time
less than that necessary for making the computation itself,
allowing the ordering party to effectively take advantage of
delegation of computation despite excess costs linked to
verification of the proof.
The Pinocchio protocol has the major disadvantage of needing
substantial computational power on the part of the executing party.
The production cost of computation proof by this protocol is in
fact greater by several orders of magnitude than that of the
computation itself.
The Pinocchio protocol is based on transcription of the function to
be evaluated in the form of an arithmetic circuit and construction
of the corresponding quadratic arithmetic program (QAP). The
computation cost rises with the number of multipliers in this
arithmetic circuit. Such a number can rapidly become large, for
example in the case where the function comprises a loop whereof
each iteration must be represented explicitly by its own operators
in the circuit. This limits the practical use of such a protocol to
evaluation of very simple functions.
This protocol was then improved via the Geppetto protocol,
presented in the publication "Craig Costello, Cedric Fournet, Jon
Howell, Markulf Kohlweiss, Benjamin Kreuter, Michael Naehrig, Bryan
Parno, and Samee Zahur, Geppetto: Versatile Verifiable Computation,
in Proceedings of the IEEE Symposium on Security and Privacy, IEEE,
18 May 2015". This novel protocol of verifiable computation
proposes cutting out the function to be evaluated into several
sub-functions, optionally used several times for evaluation of the
overall function, for example at each iteration of a loop. The
proof of evaluation of this function can then be constructed from
the less expensive proofs relative to evaluation of such
sub-functions. The total arithmetic circuit can be substantially
simplified relative to that of the Pinocchio protocol, considerably
limiting the number of multipliers to be employed. The total
production cost of the proof for the executing party is reduced
relative to that of the Pinocchio protocol. The computation cost of
a proof in terms of the Geppetto protocol continues to grow however
with the number of multipliers necessary for representation of the
function to be evaluated in the form of an arithmetic circuit. A
bootstrapping technique has been introduced to improve the
situation, but implementing this method degrades performance.
Many other derivatives of the Pinocchio protocol have been
proposed, and there are varied applications in multiple technical
fields. For example, the Cendrillon protocol presented in the
publication "Antoine Delignat-Lavaud, Cedric Fournet, Markulf
Kohlweiss and Bryan Parno, Cinderella: Turning Shabby X.509
Certificates into Elegant Anonymous Credentials with the Magic of
Verifiable Computation, in 2016 IEEE Symposium on Security and
Privacy", relates to the electronic signing of documents, and the
PhotoProof protocol, presented in the publication "Assa Naveh, Eran
Tromer, PhotoProof: cryptographic image authentication for any set
of permissible transformations, proc. IEEE Symposium on Security
& Privacy (Oakland) 2016, 255-271, IEEE, 2016" ensures that a
photograph has been modified only according to a set of admissible
transformations and has not been falsified.
Within the scope of biometric identification it is necessary to
compare a biometric fingerprint to a multitude of reference
biometric fingerprints, in a sufficiently short period to be
supported by the individual to be identified. Such comparison
between two biometric fingerprints each represented in the form of
a vector u, respectively u', of N binary integers u.sub.i or
u'.sub.i, can be made by computing the scalar product uu' between
these two vectors. The arithmetic circuit corresponding to such a
scalar product is represented in FIG. 1. It then comprises at least
N multipliers each taking on input an integer u.sub.i and the
corresponding integer u'.sub.i. Such a number of multipliers
implies a number of computations by the protocols described
hereinabove too great to be realized in real time, making such
protocols inadequate to use in terms of a biometric identification
method.
There is therefore a need for a biometric identification method for
delegating to a remote entity comparison of biometrics fingerprints
in terms of a protocol of verifiable computation, for a cost of
computation, production and verification of proof comprised, for
execution in real time.
PRESENTATION OF THE INVENTION
The present invention proposes according to a first aspect a
biometric identification method of an entity, by a biometric
identification system comprising a client device and a remote
computation device, comprising: computation of at least one
matching value between at least one biometric datum of the entity u
and at least one reference biometric datum u', by application of a
function F to said biometric data, each of said data being a vector
of N binary integers u.sub.i or u'.sub.i with 1.ltoreq.i.ltoreq.N,
each integer being coded on n bits, said function comprising a
scalar product between a biometric datum of the entity and a
reference biometric datum, said computation performing a
non-interactive, publicly verifiable computation method comprising
steps of: representation of said function in the form of an
arithmetic circuit comprising wires transporting values of the
finite prime field .sub.q, with q a prime number, and connecting
addition and multiplication operators, conversion of said
arithmetic circuit into a polynomial representation, QAP (Quadratic
Arithmetic Program) or multi-QAP, generation of a public evaluation
key and of a public verification key as a function of said
polynomial representation, obtaining by the remote computation
device of the arithmetic circuit and of the public evaluation key,
for each biometric datum of the entity, determination of at least
one matching value between said biometric datum and at least one
reference biometric datum by the remote computation device by
evaluating the arithmetic circuit having as inputs the biometric
datum of the entity and the reference biometric datum, for each
determined matching value, generation by the remote computation
device of a proof of correction of the computation execution of the
matching value, so-called generated proof, from said polynomial
representation, the public evaluation key and the result of the
evaluation of the arithmetic circuit, transmission by the remote
computation device of said matching values and of said generated
proofs to the client device, verification of said proofs received
by the client device by means of the public verification key,
identification of the entity by the client device as a function of
the matching values and of the result of said verification of
proofs, characterized in that: representation of said function
comprises encoding an integer k>1 of binary integers of a vector
of a biometric datum on at least one input wire of the circuit, and
the function F comprising at least m scalar products, m being a
divider of the length N of the biometric data vectors, if the
divider m is equal to 2 or 3, the circuit comprises at least N/km
multiplication operators connected to the input wires of the
circuit, a storage memory, and at least one addition operator, and
evaluation of the circuit iteratively comprises computation of each
of the m scalar products by means of said N/km multiplication
operators, storage of m results of computations of said scalar
products in said storage memory and summation of said results by
means of said addition operator, if the divider m is greater than
or equal to 4, the circuit comprises at least one first computation
sub-circuit of scalar product comprising N/km first multiplication
operators connected to the input wires of the circuit and a first
storage memory, and a second computation sub-circuit of scalar
product comprising N/km second multiplication operators connected
to the input wires of the circuit and a second storage memory, each
of said sub-circuit being also connected to an output of the
storage memory of the other sub-circuit, and evaluation of the
circuit iteratively comprises computation of each of the m scalar
products by using alternatively the first or the second sub-circuit
to compute the sum of the scalar product of the values of the input
wires of this sub-circuit and of the value stored in the storage
memory of the other sub-circuit.
This lets the client device delegate computation of scalar products
necessary for biometric identification of the entity, and verifies
the exactitude of computations made by the remote device, all of
this for moderate cost due to the reduced complexity of the
circuits used to represent the function of scalar product. Such
circuits in fact comprise around the order of k*m fewer multipliers
than the circuit according to the state of the art.
The verification step of said received proofs can comprise batch
verification of pairings.
In a first mode of operation, if the divider m of the length N of
the biometric data vectors is equal to 1, given an asymmetric
bilinear environment (q, G.sub.1, G.sub.2, G.sub.T, g.sub.1,
g.sub.2, e) where q is a prime number, G.sub.1, G.sub.2 and G.sub.T
three groups of order q, g.sub.1 a generator of G.sub.1, g.sub.2 a
generator of G.sub.2, and e a non-degenerate bilinear pairing e:
G.sub.1.times.G.sub.2.fwdarw.G.sub.T and the arithmetic circuit
being represented in the form of a QAP of the circuit Q=(t, V, W,
Y) of size .rho. and degree .delta., with V={vi}, W={wi}, Y={yi},
0.ltoreq.i.ltoreq..rho.,
and given I.sub.io={1, . . . , .theta.} the set of indices
corresponding to the input/output wires of the circuit and
I.sub.mid={.theta.+1, . . . , .rho.} the set of indices of
intermediate wires of the circuit not being input wires of the
circuit,
the generation step of a public evaluation key and a public
verification key comprises: generation of random variables r.sub.v,
r.sub.w, s, .alpha..sub.v, .alpha..sub.w, .alpha..sub.y, .beta.,
.gamma. in .sub.q, definition of coefficients
r.sub.y=r.sub.vr.sub.w, g.sub.v1=g.sub.1.sup.r.sup.v,
g.sub.w1=g.sub.1.sup.r.sup.w, g.sub.w2=g.sub.2.sup.r.sup.w,
g.sub.y1=g.sub.1.sup.r.sup.y and g.sub.y2=g.sub.2.sup.r.sup.y,
generation of the public evaluation key EK.sub.F equal to
(EK.sub.F1, EK.sub.F2) where
.times..times..times..times..function..di-elect
cons..times..times..function..di-elect
cons..times..times..function..di-elect
cons..times..times..alpha..function..di-elect
cons..times..times..alpha..function..di-elect
cons..times..times..function..di-elect cons..di-elect
cons..delta..times..times..beta..function..times..times..beta..function..-
times..times..beta..function..di-elect cons. ##EQU00001##
.times..times..times..times..function..di-elect cons.
##EQU00001.2## generation of the public verification key VK.sub.F
equal to (VK.sub.F1, VK.sub.F2) where:
VK.sub.F1=(g.sub.1,{g.sub.v1.sup.v.sup.i.sup.(s)}.sub.i.di-elect
cons.[1,.theta.],{g.sub.y1.sup.y.sup.i.sup.(s)}.sub.i.di-elect
cons.[1,.theta.])
VK.sub.F2=(g.sub.2,g.sub.2.sup..alpha..sup.v,g.sub.2.sup..alpha..sup.w,g.-
sub.2.sup..alpha..sup.y,g.sub.2.sup..gamma.,g.sub.2.sup..beta..gamma.,g.su-
b.y2.sup.t(s),{g.sub.w2.sup.w.sup.i.sup.(s)}.sub.i.di-elect
cons.[1,.theta.]) generation by the remote computation device of
proof of correction of the computation execution of a matching
value comprises, {c.sub.i}.sub.i.di-elect cons.[1,.rho.] being the
set of values of the circuit determined during the determination
step of the matching value: determination of a polynomial h(x) such
that p(x)=h(x)t(x) with
p(x)=(v.sub.0(x)+.SIGMA..sub.i=1.sup..rho.c.sub.iv.sub.i(x))(w.sub.0(x)+.-
SIGMA..sub.i=1.sup..rho.c.sub.iw.sub.i(x))-(y.sub.0(x)+.SIGMA..sub.i=1.sup-
..rho.c.sub.iy.sub.i(x)), computation of the generated proof
.pi.=(.pi..sub.1, .pi..sub.2) with:
.pi..times..times..function..times..times..function..times..times..functi-
on..function..times..times..alpha..function..times..times..alpha..function-
..times..times..alpha..function..times..times..beta..function..times..time-
s..beta..function..times..times..beta..function..times..pi..times..times..-
function. ##EQU00002## where: v.sub.mid(x)=.SIGMA..sub.i.di-elect
cons.I.sub.midc.sub.iv.sub.i(x),w.sub.mid(x)=.SIGMA..sub.i.di-elect
cons.I.sub.midc.sub.iw.sub.i(x) and
y.sub.mid(x)=.SIGMA..sub.i.di-elect
cons.I.sub.midc.sub.iy.sub.i(x), and, wherein said proofs received
by the client device being equal to (.pi..sub.r1, .pi..sub.r2) with
.pi..sub.r1 in the form of: (g.sub.v1.sup.V.sup.mid,
g.sub.w1.sup.W.sup.mid, g.sub.y1.sup.Y.sup.mid, g.sub.1.sup.H,
g.sub.v1.sup.V'.sup.mid, g.sub.w1.sup.W'.sup.mid,
g.sub.y1.sup.Y'.sup.mid, g.sub.1.sup.Z) and .pi..sub.r2 in the form
g.sub.w2.sup.w.sup.mid, verification of a received proof comprises
performing the following equality tests:
e(g.sub.v1.sup.v.sup.o.sup.(s)g.sub.v1.sup.v.sup.io.sup.(s)g.sub.v1.sup.V-
.sup.mid,g.sub.w2.sup.w.sup.o.sup.(s)g.sub.w2.sup.w.sup.io.sup.(s)g.sub.w2-
.sup.W.sup.mid)=e(g.sub.1.sup.H,g.sub.y2.sup.t(s))e(g.sub.y1.sup.y.sup.o.s-
up.(s)g.sub.y1.sup.y.sup.io.sup.(s)g.sub.y1.sup.Y.sup.mid,g.sub.2),
e((g.sub.v1.sup.V'.sup.mid).sup.d.sup.1(g.sub.w1.sup.W'.sup.mid).sup.d.su-
p.2(g.sub.y1.sup.Y'.sup.mid).sup.d.sup.3,g.sub.2)=e((g.sub.v1.sup.V.sup.mi-
d).sup.d.sup.1,g.sub.2.sup..alpha..sup.v)e((g.sub.w1.sup.W.sup.mid).sup.d.-
sup.2,g.sub.2.sup..alpha..sup.w)e((g.sub.y1.sup.Y.sup.mid).sup.d.sup.3,g.s-
ub.2.sup..alpha..sup.y),
e((g.sub.1.sup.Z,g.sub.2.sup..gamma.)=e(g.sub.v1.sup.V.sup.midg.sub.w1.su-
p.W.sup.midg.sub.y1.sup.Y.sup.mid,g.sub.2.sup..beta..gamma.) where
g.sub.v1.sup.v.sup.io.sup.(s)=.PI..sub.i=1.sup..theta.(g.sub.v1.sup.v.sup-
.i.sup.(s)).sup.c.sup.i,
g.sub.w1.sup.w.sup.io.sup.(s)=.PI..sub.i=1.sup..theta.(g.sub.w1.sup.w.sup-
.i.sup.(s)).sup.c.sup.i, and
g.sub.y1.sup.y.sup.io.sup.(s)=.PI..sub.i=1.sup..theta.(g.sub.y1.sup.y.sup-
.i.sup.(s)).sup.c.sup.i and (d.sub.1, d.sub.2, d.sub.3) elements of
.sub.q on .lamda. bits with .lamda. a security parameter.
Verification of the proof of computation is accelerated relative to
the Pinocchio protocol, by way of simultaneous verification of
several pairings.
In a second mode of operation, if the divider m of the length N of
the biometric data vectors is greater than or equal to 2, given an
asymmetric bilinear environment (q, G.sub.1, G.sub.2, G.sub.T,
g.sub.1, g.sub.2, e) where q is a prime number, G.sub.1, G.sub.2
and G.sub.T three groups of order q, g.sub.1 a generator of
G.sub.1, g.sub.2 a generator of G.sub.2, and e a non-degenerate
bilinear pairing e: G.sub.1.times.G.sub.2.fwdarw.G.sub.T, the
arithmetic circuit being represented in the form of a multi-QAP
Q=({B.sub.b}.sub.b.di-elect cons.[1,l],t,V,W,Y) of size .rho. and
degree .delta., with {B.sub.b}.sub.b.di-elect cons.[1,l] a set of l
banks B.sub.b of Q used in computation of the function F, and
V={vi}, W={wi}, Y={yi} with 0.ltoreq.i.ltoreq..rho., the generation
step of a public evaluation key and a public verification key
comprises: generation of random variables s,{(.alpha..sub.bv,
.alpha..sub.bw, .alpha..sub.by, .beta..sub.b,
.gamma..sub.b)}.sub.b.di-elect cons.[1,l], r.sub.v, r.sub.w in
.sub.q, definition of the following coefficients:
r.sub.y=r.sub.br.sub.w, g.sub.v1=g.sub.1.sup.r.sup.v,
g.sub.v2=g.sub.2.sup.r.sup.v, g.sub.w1=g.sub.1.sup.r.sup.w,
g.sub.w2.sup.r.sup.w=g.sub.2.sup.r.sup.w,
g.sub.y1=g.sub.1.sup.r.sup.y and g.sub.y2=g.sub.2.sup.r.sup.y,
generation of the public evaluation key EK.sub.F equal to:
({EK.sub.Fb}.sub.b.di-elect
cons.[1,l],{g.sub.1.sup.s.sup.i}.sub.i.di-elect
cons.[1,.delta.],g.sub.v1.sup.t(s),g.sub.w1.sup.t(s),g.sub.y1.sup.t(s),g.-
sub.v2.sup.t(s),g.sub.w2.sup.t(s),g.sub.y2.sup.t(s))
where each public bank key EK.sub.Fb is equal to (EK.sub.Fb1,
EK.sub.Fb2) with:
.times..times..times..times..function..times..times..function..times..tim-
es..function..times..times..alpha..function..times..times..alpha..function-
..times..times..alpha..function..times..times..beta..function..times..time-
s..beta..function..times..times..beta..function..di-elect
cons..times..times..alpha..function..times..times..alpha..function..times-
..times..alpha..function..times..times..beta..function..times..times..beta-
..function..times..times..beta..function. ##EQU00003##
.times..times..times..times..times..function..di-elect
cons..beta..times..times..alpha..function. ##EQU00003.2##
generation of the public verification key VK.sub.F equal to:
({VK.sub.Fb}.sub.b.di-elect cons.[1,l], g.sub.1, g.sub.2,
g.sub.y2.sup.t(s)) where each public bank key VK.sub.Fb is equal to
(g.sub.2.sup..alpha..sup.bv, g.sub.2.sup..alpha..sup.bw,
g.sub.2.sup..alpha..sup.by, g.sub.2.sup..gamma..sup.b,
g.sub.2.sup..beta..sup.b.sup..gamma..sup.b) determination of a
matching value comprises, function F being divided into .omega.
sub-functions F.sub.1, . . . , F.sub..omega. and .sigma.=((f.sub.l,
(T.sub.l1, . . . , T.sub.ll))).sub.l.di-elect cons.[1,L] being a
scheduling of length L with f.sub.l .di-elect cons.{1, . . . ,
.omega.}, evaluation of each sub-function F.sub..omega. from the
biometric data of the entity and the reference biometric data and
determination of the values i of the circuit, generation by the
remote computation device of proof of correction of the computation
execution of a matching value comprises, for each l={1, . . . , L}:
for each bank B.sub.b such that b.di-elect cons..LAMBDA., with
.LAMBDA.[1,l] the set of indices b.di-elect cons.[1,l] such that
T.sub.lb.noteq.0,.GAMMA.=U.sub.b.di-elect cons..LAMBDA.B.sub.b,
{c.sub.j}.sub.j.di-elect cons.B.sub.b an instance of the bank
B.sub.b, .DELTA.={c.sub.i}.sub.i.di-elect cons..GAMMA. the set of
values of .GAMMA.: generation of pledging random variables in
.sub.q: o.sub.b=(o.sub.bv, o.sub.bw, o.sub.by), computation of a
digest D.sub.b equal to (D.sub.b1,D.sub.b2) from the instance of
the bank of variables B.sub.b:
B.sub.b.sup.(T.sup.lb.sup.)={c.sub.i.di-elect
cons..DELTA.}.sub.i.di-elect cons.B.sub.b and pledging random
variables o.sub.b and such that: if the bank B.sub.b is an
input/output bank:
D.sub.b1=(g.sub.v1.sup.v.sup.(b).sup.(s),g.sub.y1.sup.y.sup.(b).sup.(s))
and D.sub.b2=(g.sub.w2.sup.w.sup.(b).sup.(s)), if the bank B.sub.b
is not an input/output bank:
.times..times..times..times..function..times..times..function..times..tim-
es..function..times..times..alpha..function..times..times..alpha..function-
..times..times..alpha..function..times..times..beta..function..times..time-
s..beta..function..times..times..beta..function. ##EQU00004##
.times..times..times..times..function. ##EQU00004.2## with:
v.sup.(b)(s)=.SIGMA..sub.i.di-elect
cons.B.sub.bc.sub.iv.sub.i(s)+o.sub.bvt(s),
w.sup.(b)(s)=.SIGMA..sub.i.di-elect
cons.B.sub.bc.sub.iw.sub.i(s)+o.sub.bwt(s),
y.sup.(b)(s)=.SIGMA..sub.i.di-elect
cons.B.sub.bc.sub.iy.sub.i(s)+o.sub.byt(s), determination of a
polynomial h.sup.(l)(x) such that p.sup.(l)(x)=h.sup.(l)(x)t(x)
with p.sup.(l)(x)=(v.sub.0(x)+.SIGMA..sub.i.di-elect
cons..GAMMA.c.sub.iv.sub.j(x)+.SIGMA..sub.b.di-elect
cons..LAMBDA.o.sub.bvt(x))(w.sub.0(x)+.SIGMA..sub.i.di-elect
cons..GAMMA.c.sub.iw.sub.j(x)+.SIGMA..sub.b.di-elect
cons..LAMBDA.o.sub.bwt(x))-(y.sub.0(x)+.SIGMA..sub.i.di-elect
cons..GAMMA.c.sub.iy.sub.j(x)+.SIGMA..sub.b.di-elect
cons..LAMBDA.o.sub.byt(x)) computation of a proof element
.pi..sup.(l) equal to g.sub.1.sup.h.sup.(l).sup.(s), and, wherein
said proofs received by the client device being of the form
D.sub.1.sup.(1), . . . , D.sub.l.sup.(1), .pi..sup.(1), . . . ,
D.sub.1.sup.(L), . . . , D.sub.l.sup.(L), .pi..sup.(L) where for
all l.di-elect cons.{1, . . . , L} and b.di-elect cons.{1, . . .
,l}:
.times..times..times..times..times..times..times..times.'.times..times..t-
imes..times.'.times..times..times..times.'.times..times..times..times.
##EQU00005## and .pi..sup.(l)=g.sub.1.sup.H.sup.(l), verification
of a received proof (E8) comprises: verification of Ll digests, for
l.di-elect cons.{1, . . . , L} and b.di-elect cons.{1, . . . , l}
comprising performing the following equality tests:
.function..times..times.'.times..times..alpha..function..times..times.
##EQU00006##
.function..times..times.'.function..alpha..function..times..times.
##EQU00006.2##
.function..times..times.'.times..times..alpha..function..times..times.
##EQU00006.3##
.function..gamma..function..times..times..times..times..times..times..bet-
a..gamma. ##EQU00006.4## verification of L proofs comprising for
l.di-elect cons.{1, . . . , L} performing the following equality
test:
.function. .times..times..times.
.times..times..times..function..times..times..function..function.
.times..times..times. ##EQU00007##
Verification of the proof of computation is accelerated relative to
the Geppetto protocol, by way of simultaneous verification of
several pairings.
In a third mode of operation, if the divider m of the length N of
the biometric data vectors is greater than or equal to 2, given an
asymmetric bilinear environment (q, G.sub.1, G.sub.2, G.sub.T,
g.sub.1, g.sub.2, e) where q is a prime number G.sub.1, G.sub.2 and
G.sub.T three groups of order q, g.sub.1 a generator of G.sub.1,
g.sub.2 a generator of G.sub.2, and e a non-degenerate bilinear
pairing e: G.sub.1.times.G.sub.2.fwdarw.G.sub.T, the arithmetic
circuit being represented in the form of a multi-QAP
Q=({B.sub.b}.sub.b.di-elect cons.[1,l], t,V,W,Y) of size .rho. and
degree .delta., with {B.sub.b}.sub.b.di-elect cons.[1,l] a set of l
banks B.sub.b of Q used in computation of the function F, and
V={vi}, W={wi}, Y={yi} with 0.ltoreq.i.ltoreq..rho., generation
step of a public evaluation key and a public verification key
comprises: generation of random variables s, {(.alpha..sub.bv,
.alpha..sub.bw, .alpha..sub.by, .beta..sub.b,
.gamma..sub.b)}.sub.b.di-elect cons.[1,l], r.sub.v, r.sub.w in
.sub.q, definition of the following coefficients:
r.sub.y=r.sub.vr.sub.w, g.sub.v1=g.sub.1.sup.r.sup.v,
g.sub.v2=g.sub.2.sup.r.sup.v, g.sub.w1=g.sub.1.sup.r.sup.w,
g.sub.w2=g.sub.2.sup.r.sup.w, g.sub.y1=g.sub.1.sup.r.sup.y and
g.sub.y2=g.sub.2.sup.r.sup.y, generation of the public evaluation
key EK.sub.F equal to: ({EK.sub.Fb}.sub.b.di-elect
cons.[1,l],{g.sub.1.sup.s.sup.i}.sub.i.di-elect
cons.[1,.delta.],g.sub.v1.sup.t(s),g.sub.w1.sup.t(s),g.sub.y1.sup.t(s),g.-
sub.v2.sup.t(s),g.sub.w2.sup.t(s),g.sub.y2.sup.t(s)) where each
public bank key EK.sub.Fb is equal to (EK.sub.Fb1, EK.sub.Fb2)
with:
.times..times..times..times..function..times..times..function..times..tim-
es..function..times..times..alpha..function..times..times..alpha..times..f-
unction..times..times..alpha..function..times..times..beta..function..time-
s..times..beta..function..times..times..beta..function..di-elect
cons..times..times..alpha..function..times..times..alpha.
.function..times..times..alpha..function..times..times..beta..function..t-
imes..times..beta..function..times..times..beta..function.
##EQU00008## .times..times..times..times..times..function..di-elect
cons..times..times..alpha..function. ##EQU00008.2## generation of
the public verification key VK.sub.F equal to:
({VK.sub.Fb}.sub.b.di-elect cons.[1,l], g.sub.1, g.sub.2,
g.sub.y2.sup.t(s)) where each public bank key VK.sub.Fb is equal to
(g.sub.2.sup..alpha..sup.bv, g.sub.2.sup..alpha..sup.bw,
g.sub.2.sup..alpha..sup.by, g.sub.2.sup..gamma..sup.b,
g.sub.2.sup..beta..sup.b.sup..gamma..sup.b), determination of a
matching value comprises, function F being divided into .omega.
sub-functions F.sub.1, . . . , F.sub..omega. and .sigma.=((f.sub.l,
(T.sub.l1, . . . , T.sub.ll))).sub.l.di-elect cons.[1,L] being a
scheduling of length L with f.sub.l.di-elect cons.{1, . . . ,
.omega.}, evaluation of each sub-function F.sub..omega. from the
biometric data of the entity and the reference biometric data and
determination of the values of the circuit, generation by the
remote computation device of proof of correction of the computation
execution of a matching value comprises, for each l={1, . . . , L}:
for each bank B.sub.b such that b.di-elect cons..LAMBDA., with
.LAMBDA.[1,l] the set of indices b.di-elect cons.[1,l] such that
T.sub.lb.noteq.0, .GAMMA.=U.sub.b.di-elect cons..LAMBDA.B.sub.b,
{c.sub.j}.sub.j.di-elect cons..sub.B an instance of the bank
B.sub.b, .DELTA.={c.sub.i}.sub.i.di-elect cons..GAMMA. the set of
values of r: generation of pledging random variables in .sub.q:
o.sub.b=(o.sub.bv, o.sub.bw, o.sub.by), computation of a digest
D.sub.b equal to (D.sub.b1,D.sub.b2) from the instance of the bank
of variables B.sub.b: B.sub.b.sup.(T.sup.lb.sup.)={c.sub.i.di-elect
cons..DELTA.}.sub.i.di-elect cons.B.sub.b and pledging random
variables o.sub.b and such that: if the bank B.sub.b is an
input/output bank: D.sub.b1=(g.sub.v1.sup.v.sup.(b).sup.(s),
g.sub.y1.sup.v.sup.(b).sup.(s)) and
D.sub.b2=(g.sub.w2.sup.w.sup.(b).sup.(s)). if the bank B.sub.b is
not an input/output bank:
.times..times..times..times..function..times..times..function..times..tim-
es..function..times..times..alpha..function..times..times..alpha..function-
..times..times..alpha..function..times..times..beta..function..times..time-
s..beta..function..times..times..beta..function. ##EQU00009##
.times..times..times..times..function. ##EQU00009.2## with:
v.sup.(b)(s)=.SIGMA..sub.i.di-elect
cons.B.sub.bc.sub.iv.sub.i(s)+o.sub.bvt(s),
w.sup.(b)(s)=.SIGMA..sub.i.di-elect
cons.B.sub.bc.sub.iw.sub.i(s)+o.sub.bwt(s),
y.sup.(b)(s)=.SIGMA..sub.i.di-elect
cons.B.sub.bc.sub.iy.sub.i(s)+o.sub.byt(s), determination of a
polynomial h.sup.(l)(x) such that p.sup.(l)(x)=h.sup.(l)(x)t(x)
with p.sup.(l)(x)=(v.sub.0(x)+.SIGMA..sub.i.di-elect
cons..GAMMA.c.sub.iv.sub.j(x)+.SIGMA..sub.b.di-elect
cons..LAMBDA.o.sub.bvt(x))(w.sub.0(x)+.SIGMA..sub.i.di-elect
cons..GAMMA.c.sub.iw.sub.j(x)+.SIGMA..sub.b.di-elect
cons..LAMBDA.o.sub.bwt(x))-(y.sub.0(x)+.SIGMA..sub.i.di-elect
cons..GAMMA.c.sub.iy.sub.j(x)+.SIGMA..sub.b.di-elect
cons..LAMBDA.o.sub.byt(x)) computation of a proof element
.pi..sup.(l) equal to g.sub.1.sup.h.sup.(l).sup.(s), and, wherein
said proofs received by the client device being of the form
D.sub.1.sup.(1), . . . , D.sub.l.sup.(1), .pi..sup.(1), . . . ,
D.sub.1.sup.(L), . . . , D.sub.l.sup.(L), .pi..sup.(L) where for
all l.di-elect cons.{1, . . . , L} and b.di-elect cons.{1, . . .
,l}:
.times..times..times..times..times..times..times..times.'.times..times..t-
imes..times.'.times..times..times..times.'.times..times..times..times.
##EQU00010## and .pi..sup.(l)=g.sub.1.sup.H.sup.(l), verification
of a received proof comprises, given a correction parameter
.lamda.: selection of a random vector (d.sub.1, . . . , d.sub.3l)
of elements of size .lamda., batch verification of Ll digests
comprising performing l times the following equality tests, for
b.di-elect cons.{1, . . . , l}:
.function..times..times..times.'.times..times..alpha..function..times..ti-
mes..times.'.times..times..alpha..function..times..times..times.'.times..t-
imes..alpha..function..times..times..times..times..times..times..times..ti-
mes..times. ##EQU00011##
.times..function..times..gamma..function..times..times..times..times..tim-
es..times..times..beta..gamma. ##EQU00011.2## batch verification of
L proofs comprising performing the following equality test:
.times..times..function. .times..times..times..times..function.
.times..times..times..times..function..function..times..times..times..tim-
es..function..function..times..times. .times..times..times..times.
##EQU00012##
This makes verification of a proof even faster due to batch
verification of digests of the proof.
Identification of the entity can comprise comparison of the
matching values with a predetermined threshold.
Function F can comprise comparison of the result of the scalar
product between said biometric data of the entity and said
reference biometric data with a predetermined threshold.
Such comparison to a threshold decides if the compared biometric
data are sufficiently close to conclude successful identification
of the entity to be identified.
Encoding of k binary integers u.sub.i or u'.sub.i on an input wire
of an j.sup.th multiplication operator, 1.ltoreq.j.ltoreq.N/k, is
equal to
.times..times..function. ##EQU00013## ##EQU00013.2##
.times..times..function.' ##EQU00013.3## with 1.ltoreq.z.ltoreq.m,
and is given by the formula:
.times..function..times..times..times..times. ##EQU00014##
with .di-elect cons..sub.1, . . . , .di-elect cons..sub.k
predetermined integers.
Several integers of the biometric data can be encoded on each input
wire, reducing the number of multipliers necessary for computation
of the scalar product between a biometric datum of the entity and a
reference biometric datum.
According to a second aspect, the invention relates to a computer
program product comprising code instructions for execution of a
method according to the first aspect when this program is executed
by a processor.
According to a third aspect, the invention relates to a biometric
identification system comprising a client device and a remote
computation device characterized in that said client device and
said remote computation device each comprise a processor, an
interface and a memory for performing the steps of the
identification method according to the first aspect.
Such computer program product and system have the same advantages
as those mentioned for the method according to the first
aspect.
PRESENTATION OF THE FIGURES
Other characteristics and advantages of the present invention will
emerge from the following description of a preferred embodiment.
This description will be given in reference to the appended
drawings, in which:
FIG. 1 illustrates an arithmetic circuit corresponding to naive
implementation of a scalar product for biometric identification of
an entity;
FIG. 2 illustrates an identification system according to an
embodiment of the invention;
FIG. 3 is a diagram showing an implementation of an identification
method according to the invention;
FIGS. 4a and 4b illustrate an arithmetic circuit for biometric
identification of an entity according to a first mode of operation
of the invention;
FIGS. 5a and 5b illustrate an example implementation of the circuit
of FIG. 4 for N=3000, n=8, q=256, k=3;
FIGS. 6a to 6c illustrate a arithmetic circuit for biometric
identification of an entity according to a second mode of operation
of the invention;
FIGS. 7a and 7b illustrate an arithmetic circuit for biometric
identification of an entity according to a third mode of operation
of the invention;
DETAILED DESCRIPTION
The present invention relates to implementing a biometric
identification method of an entity 1 by an identification system 2
comprising a client device 3 and a remote computation device 4
capable of being connected together by a communications network 5,
as represented in FIG. 2. Such an entity can be an individual or
else an object having biometric data such as a biometric
passport.
The client device and the remote computation device can each
comprise a random access memory and internal storage means such as
rewritable non-volatile memory (flash memory or EEPROM memory) and
processing means comprising a processor. They can also comprise an
interface for dialoguing with each other, of wired type such as an
Ethernet link, or wireless such as a Wifi or Bluetooth
connection.
The aim of this method carried out is to allow the device to
delegate to the remote computation device the computations
necessary for biometric identification of the entity to be
identified, so that the computations made by the remote computation
device are publicly verifiable, all this happening over a
sufficiently short period to be acceptable in terms of an
identification method.
For conducting such biometric identification the client device
acquires at least one biometric datum of the entity to be
identified u. To identify the entity, this at least one biometric
datum u must be compared to one or more reference biometric data
u', stored in advance.
By way of example, such biometric data can be fingerprints, DNA,
voice or even iris images or venous networks. Each of these
biometric data is a vector of N binary integers u.sub.i or u'.sub.i
with 1.ltoreq.i.ltoreq.N. Each integer u.sub.i or u'.sub.i is coded
on n bits. For example in the case of a face biometric datum, there
can typically be N=3000 and n=8.
The client device can comprise or be connected to a device for
capturing such biometric data, such as a fingerprint reader, a
microphone, or an iris-imaging device. This capture device can be
employed to acquire the biometric datum u acquired for the entity
1. The reference biometric data u' can be stored in the storage
means of the client device or of the remote computation device.
The identification method can comprise the steps described
hereinbelow in reference to FIG. 3.
First of all at least one matching value can be computed F1 between
at least one biometric datum of the entity u and at least one
reference biometric datum u', by application of a function F,
so-called correlation function, to said biometric data. Function F
comprises a scalar product between a biometric datum of the entity
and a reference biometric datum. Such a scalar product in fact
computes a score S=.SIGMA..sub.j=1.sup.N(u.sub.ju.sub.j') which is
all the higher since the data compared are similar. Such a score
can be used as matching value. To determine if an attempt at
identification has succeeded, the matching values can be compared
to a predetermined threshold T. Alternatively, function F can
comprise the comparison of the result of the scalar product between
the biometric data of the entity u and the reference biometric data
u', i.e. of the score S, to this predetermined threshold T.
Computation of the matching value can comprise computation of the
value (S-T) and the matching value coming from such computation can
be a sign bit for example taking the value 1 if S-T>0, the value
0 if not.
To ensure the quality of the result obtained, computation of such a
matching value F1 employs a non-interactive, publicly verifiable
computation method. Such a method generally being divided into
three phases: representation of the function F to be evaluated in
the form of a system constraints represented in the form of an
arithmetic circuit. transformation of this arithmetic circuit into
polynomial representation called "Quadratic Arithmetic Programs"
(abbreviated to QAP below). generation of a proof of correction of
the computation execution from the QAP.
The non-interactive, publicly verifiable computation method carried
out more precisely first comprises a representation step E1 of said
function F in the form of an arithmetic circuit. Such an arithmetic
circuit comprises wires transporting values of the finite prime
field Zq, with q a prime number, and connecting addition and
multiplication operators. Typically the size q of the values of the
circuit wires can be equal to 256 bits.
The arithmetic circuit is then converted E2 into a polynomial
representation, QAP ("Quadratic Arithmetic Program") or multi-QAP.
Such representations and the way to attain them from an arithmetic
circuit are described in more detail in the publications cited
hereinabove on the existing Pinocchio and Geppetto protocols.
Next a public evaluation key and a public verification key are
generated E3 as a function of said polynomial representation. The
remote computation device then obtains E4 the arithmetic circuit
and the public evaluation key.
The representation steps in the form of an arithmetic circuit E1,
conversion into a polynomial representation E2 and generation of
keys E3 can be conducted by the client device itself. Alternatively
these steps can be delegated to a trusted third party. Since such
steps are independent of the value of the biometric data to be
compared, they can be conducted once only, prior to comparisons of
biometric data described hereinbelow, and do not need to be
repeated as long as the format of the biometric data to be compared
does not change.
For each biometric datum of the entity at least one matching value
between said biometric datum and at least one reference biometric
datum is then determined E5 by the remote computation device by
evaluating the arithmetic circuit with the biometric data of the
entity and the reference biometric datum as inputs.
For each determined matching value the remote computation device
generates E6 a proof of correction of the computation execution of
the matching value, so-called generated proof, from said polynomial
representation, the public evaluation key and the result of
evaluation of the arithmetic circuit. It then transmits E7 the
matching values and said generated proofs to the client device.
The latter verifies E8 the received proof by means of the public
verification key. The verification step of said received proofs E8
can comprise or not batch verification of pairings.
Finally the entity is identified F2 by the client device as a
function of the matching values and the result of said verification
of proofs.
The integers u.sub.i and u.sub.i' constituting the data of the
entity and the reference data are usually encoded on a number of
bits n far less than the size q of the values of wires of the
circuit. By way of example the number of bits n can be equal to 8
bits and the size q can be equal to 256 bits. To limit the number
of multipliers necessary for representation of the function F in
the form of an arithmetic circuit, several integers u respectively
u.sub.i' are encoded on each input wire of the arithmetic circuit.
Representation of said function E1 comprises encoding an integer
k>1 of binary integers of a vector of a biometric datum on at
least one input wire of the circuit. In practice, encoding
E.sub.k.sup.((j-1)k+1)(u) or E.sub.k.sup.((j-1)k+1)(u') of k binary
integers u.sub.i or u'.sub.i on an input wire of a j.sup.th
multiplication operator, 1.ltoreq.j.ltoreq.N/k, can be defined by
the formula:
.times..function..times..times..times..times. ##EQU00015## with
.di-elect cons..sub.1, . . . , .di-elect cons..sub.k predetermined
integers.)
A multiplier having on input E.sub.k.sup.((j-1)k+1)(u) and
E.sub.k.sup.((j-1)k+1)(u') has on its output wire the product of
encodings of successive k binary integers u.sub.i or u'.sub.i coded
on its input wires. This product is noted
E.sub.uu',k.sup.((j-1)k+1)=E.sub.k.sup.((j-1)k+1)(u).
E.sub.k.sup.((j-1)k+1)(u'). By way of example, for j=1, there is:
E.sub.uu',k.sup.(1)=2.sup.2.di-elect
cons..sup.1u.sub.1u.sub.1'+2.sup..di-elect
cons..sup.2(u.sub.1u.sub.2'+u.sub.2u.sub.1')+2.sup.2.di-elect
cons..sup.2u.sub.2u.sub.2'+ . . . +2.sup.2.di-elect
cons..sup.ku.sub.ku.sub.k'
To further reduce the number of multipliers of the arithmetic
circuit, the method as carried out also proposes splitting
computation of the scalar product of the biometric datum of the
entity u and of the reference biometric datum u' of lengths N into
several computations of scalar products of vectors of lesser size
coming from splitting of the vectors u and u'. The combination of
the results of these scalar products produces the score S
corresponding to the result of the scalar product of u and u'.
For this, function F comprising at least m scalar products,
function F can be decomposed into at least m occurrences of
sub-functions, m being a divider m of the length N of the biometric
data vectors. Only the split sub-functions are represented by their
own sub-circuit in the arithmetic circuit, reducing the number of
multipliers of the circuit. To combine decomposition of the scalar
product of u and u' into m scalar sub-products, and coding of k
integers on each input wire of the circuit, it is possible to
select k such that k divides m. The scalar product of u and u' can
be decomposed into m scalar sub-products of vectors of length N/km.
The sum of the results of these m scalar products produces an
encoded score {tilde over (S)} defined by the following
formula:
.times..times.'.times..times..times. .times..times.'' ##EQU00016##
and of the following form if the expression hereinabove is deployed
and if the terms are gathered by power of 2: {tilde over
(S)}=2.sup.2.di-elect cons..sup.1a.sub.1+2.sup.2.di-elect
cons..sup.2a.sub.2'+2.sup.2.di-elect
cons..sup.2a.sub.2+2.sup.2.di-elect cons..sup.3a.sub.3'+ . . .
+2.sup.2.di-elect cons..sup.ka.sub.k with a.sub.i the terms at
2.sup.2.di-elect cons..sup.i gathering the products of integers
u.sub.i and u'.sub.i of the same indices useful for computing the
scalar product and u and u', and with a.sub.i' the terms at
2.sup..di-elect cons..sup.2, . . . , 2.sup..di-elect cons..sup.k
gathering the remaining products, cross products of integers
u.sub.i and u'.sub.j of different indices not useful for
computation of the scalar product of u and u'.
To extract the score S from its encoded version {tilde over (S)},
it is possible to extract the k sub-terms corresponding to the
coefficients 2.sup.2.di-elect cons..sup.1, . . . , 2.sup.2.di-elect
cons..sup.k (i.e., the elements a.sub.1, a.sub.2, . . . , a.sub.k),
then add them.
The paragraphs hereinbelow present the specific features of the
method for different ranges of value of the divider m. m can be
determined by making a compromise between the computational powers
of the client device and the remote computation device as well
especially as memories.
In a first mode of operation in which the divider m is equal to 1,
function F can be put in the form of the circuit represented in
FIGS. 4a and 4b. Such a circuit comprises N/k input multipliers
each taking on input E.sub.k.sup.((j-1)k+1)(u) and
E.sub.k.sup.((j-1)k+1)(u') with 1.ltoreq.j.ltoreq.N/k. In FIGS. 4a
and 4b these multipliers are noted r.sub.j and the input wires are
numbered from 1 to 2N/k. The N/k output wires of the multipliers
carry the values E.sub.uu',k.sup.((j-1)k+i) whereof the sum is
equal to the encoded score {tilde over (S)} described hereinabove.
The circuit comprises an additional output multiplier, numbered
r.sub.N/k+1, necessary for conversion of the circuit in the form of
QAP. This multiplier multiplies the output of the scalar product by
1 so as not to modify the encoded score {tilde over (S)}. Finally,
a split gate is employed to extract the coefficients a.sub.i with
1.ltoreq.i.ltoreq.k corresponding to the terms of the scalar
product uu'. The summation of these terms first split from the
encoded score then reconstituted by multiplication by powers of two
and added produces the score S. FIGS. 4a and 4b correspond to an
execution in which the score is compared to the predetermined
threshold T. The value (S-T) is computed and provided on input with
a second split gate to extract the sign bit indicating the result
of the comparison. More precisely, hereinbelow split gate means an
arithmetic gate which splits an integer bit by bit. Given an
integer a.di-elect cons..sub.q, as is known to keep on .tau. bits,
the split gate contains an input wire (containing the integer a)
and .tau. output wires. In terms of elementary arithmetic
constraints, its definition is given for example in the cited
article Pinocchio, paragraph 3.2. It is recalled here by way of
indication. It is clear that c.sub.0 is the input wire and c.sub.1,
. . . , c.sub..tau. the output wires. The arithmetic circuit of the
gate so-called "split gate" is defined as follows: concatenation of
the bits on output is equal to the input
.tau..times. ##EQU00017## each output wire contains a Boolean value
(0 or 1): .A-inverted.i.di-elect cons.{1, . . .
,.tau.}:c.sub.j(1-c.sub.j)=0 When a "split gate" gate is used
within a circuit, the integer .tau. is determined as an achieved
upper limit given the size of the circuit inputs and all the
arithmetic gates located between the inputs and the split gate.
An example of implementation is represented in FIGS. 5a and 5b for
N=3000, n=8, q=256, k=3. In such an mode of operation, the
following values can be used for the parameters
.epsilon..sub.i:.epsilon..sub.1=0, .epsilon..sub.2=26,
.epsilon..sub.3=78.
In a second mode of operation in which the divider m is equal to 2
or 3, function F can be decomposed into a function F1 computing a
scalar sub-product between two vectors of size N/km, to be used m
times, and a function F2 computing the sum of m values,
corresponding to a coded score, and performing extraction of the
corresponding score equal to the preferred scalar product.
As represented in FIGS. 6a to 6c, the corresponding circuit
comprises at least N/km multiplication operators connected to the
input wires of the circuit, a storage memory, and at least one
addition operator, and evaluation of the circuit iteratively
comprises computation of each of the m scalar products by means of
said N/km multiplication operators, storage of the m results of
computations of said scalar products in said storage memory and
summation of said results by means of said addition operator.
With F.sub.1 and F.sub.2 defined as such, evaluation of function F
corresponds to m applications of function F.sub.1 followed by
application of function F.sub.2. The circuit represented in FIGS.
6a to 6c comprises N/km input multipliers each taking on input,
during the iteration z of the function F1,
.times..times..function. ##EQU00018## ##EQU00018.2##
.times..times..function.' ##EQU00018.3## with
1.ltoreq.j.ltoreq.N/km. In FIGS. 6a to 6c these multipliers are
noted r.sub.j and the input wires are numbered from 1 to 2
N/km.
The m applications of function F.sub.1 compute the coded sub-scores
{tilde over (S)}.sub.z, for z.di-elect cons.{1, . . . , m}:
.times.' ##EQU00019##
By way of example, for z=1, there is:
.times..times.'.times..times..times. .times..times.''
##EQU00020##
During the iteration z the N/km output wires of the multipliers
thus carry the values
'.times..times. ##EQU00021## whereof the sum is equal to the
encoded score {tilde over (S)}.sub.z described hereinabove. The
circuit comprises an additional output multiplier, numbered
r.sub.N/km+1, necessary for conversion of the circuit into the form
of QAP. The m coded sub-scores {tilde over (S)}.sub.z noted {tilde
over (S)}.sub.z.sup.(out) in FIGS. 6a to 6c are stored in the
storage memory corresponding to the m bus ("Bus Bank") of the
verifiable computation method. Evaluation of function F2, with on
input the m values coming from the buses, noted {tilde over
(S)}.sub.z.sup.(in) in FIGS. 6a to 6c, produces the coded score S
described hereinabove.
As in the case of the circuit in FIGS. 4a and 4b, a first split
gate recovers the terms a.sub.i with 1.ltoreq.i.ltoreq.k
corresponding to the terms of the scalar product uu'. The summation
of these terms produces the score S which is then compared to the
predetermined threshold T.
According to a variant not represented, function F1 can comprise
decoding of the coded sub-score {tilde over (S)}.sub.z obtained
during its evaluation into a sub-score S.sub.z. Such decoding can
be done similarly to the decoding of the coded score {tilde over
(S)} presented hereinabove. Function F2 comprises only the
summation of the sub-scores S.sub.z to obtain the score S
corresponding to the scalar product uu', according to the
formula:
.times..times..times..times..times.'.times..times.'.times.'.times.
##EQU00022##
In a third mode of operation in which the divider m is greater than
or equal to 4, it is possible to decompose function F into a
function F.sub.1 and a function F.sub.2, alternatively use and m
times the total, which each take on input two vectors of size N/km
and a sub-score, and on output return an updated sub-score defined
as the sum of the sub-score given on input with the result of the
scalar product of the vectors provided on input; and a function
F.sub.3 which decodes a coded score {tilde over (S)} into a score
S.
As represented in FIGS. 7a and 7b, the circuit comprises at least
one first computation sub-circuit of scalar product comprising N/km
first multiplication operators connected to the input wires of the
circuit, for evaluation of function F1 for z odd, and a first
storage memory, and a second computation sub-circuit of scalar
product comprising N/km second multiplication operators connected
to the input wires of the circuit, for evaluation of function F2
for z even, and a second storage memory, each of said sub-circuit
being also connected to an output of the storage memory of the
other sub-circuit, and evaluation of the circuit iteratively
comprises computation of each of the m scalar products by using
alternatively the first or the second sub-circuit to compute the
sum of the scalar product of the values of the input wires of this
sub-circuit and of the value stored in the storage memory of the
other sub-circuit.
With F.sub.1, F.sub.2 and F.sub.3 defined as such, evaluation of
the function F then corresponds to m applications alternatively of
function F.sub.1 and function F.sub.2 followed by application of
function F.sub.3.
The circuit represented in FIGS. 7a and 7b comprises two sets of
N/km input multipliers. During the iteration z of the function F1
or F2, the N/km multipliers of the relevant sub-circuit each take
on input,
.times..times..function. ##EQU00023## ##EQU00023.2##
.times..times..function.' ##EQU00023.3## with
1.ltoreq.j.ltoreq.N/km. In FIGS. 7a and 7b these multipliers are
noted r.sub.j with respectively 1.ltoreq.j.ltoreq.N/km and
N/km+2.ltoreq.j.ltoreq.2N/km+1. The input wires are numbered
respectively from 1 to 2N/km and from 3N/km+3 to 5N/km+2.
The m applications of functions F.sub.1 and F.sub.2 compute the
coded sub-scores {tilde over (S)}.sub.z, for z.di-elect cons.{1, .
. . , m}:
.zeta..times..times.'.zeta. ##EQU00024##
The N/km output wires of the multipliers of a sub-circuit during
the iteration z carry the values
'.times..times. ##EQU00025## whereof the sum, added to the coded
sub-score of the preceding iteration {tilde over (S)}.sub.z-1, is
equal to the encoded score {tilde over (S)}.sub.z described
hereinabove. The sub-score {tilde over (S)}.sub.0 can be also
initialized at 0 during the first iteration. The coded score {tilde
over (S)} is constructed iteratively, each iteration adding to the
sub-score coming from the preceding iteration the result of the
current scalar sub-product
.SIGMA..sub.j=1.sup.N/kmE.sub.uu',k.sup.((z-1)(N/m)+(j-1)k+1).
The circuit comprises an additional output multiplier for each
sub-circuit, numbered r.sub.N/km+1, r.sub.2N/km+2 necessary for
conversion of the circuit in the form of QAP. On completion of its
evaluation each sub-circuit stores the coded sub-scores {tilde over
(S)}.sub.z computed in the storage memory corresponding to its bus
(Bus Bank) r2.sub.N/km+3, r.sub.2N/km+219 in terms of the
verifiable computation method.
On completion of m iterations of functions F1 and F2, the coded
score g is thus stored in one of the two storage memories. In the
example of FIGS. 7a and 7b, it is supposed that m is odd. The coded
score {tilde over (S)} is stored in the first storage memory.
During evaluation of the function F3, with the coded score {tilde
over (S)} on input a first split gate recovers the terms a.sub.i
with 1.ltoreq.i.ltoreq.k corresponding to the terms of the scalar
product uu'. The summation of these terms produces the score S
which is then compared to the predetermined threshold T.
Within the scope of the method described hereinabove the operations
to be carried out for generation of the evaluation and verification
public keys, and for generation and verification of computation
proof can be derived from existing verifiable computation protocols
such as Pinocchio, when m=1, and Geppetto, when m>1. The
paragraphs hereinbelow describe these operations in more detail as
a function of the value of the divider m.
It is to be understood that the embodiment to be described is a
particularly advantageous embodiment which is not limiting. The
skilled person can use other ways to perform generation of the
evaluation and verification public keys, generation and
verification of computation proof, and derive said operations from
other verifiable existing computation protocols.
Case m=1: In the first mode of operation where m=1, an asymmetric
bilinear environment (q, G.sub.1, G.sub.2, G.sub.T, g.sub.1,
g.sub.2, e) is defined with q a prime number, G.sub.1, G.sub.2 and
G.sub.T three groups of order q, g.sub.1 a generator of G.sub.1,
g.sub.2 a generator of G.sub.2, and e a non-degenerate bilinear
pairing e: G.sub.1.times.G.sub.2.fwdarw.G.sub.T.
The arithmetic circuit can be represented in the form of a
polynomial representation of the circuit Q=(t,V,W,Y) of size .rho.
and degree .delta., with V={vi}, W={wi}, Y={yi},
0.ltoreq.i.ltoreq..rho..
The following are noted hereinbelow:
I.sub.io={1, . . . , .theta.} the set of indices corresponding to
the input/output wires of the circuit,
I.sub.mid={.theta.+1, . . . , .rho.} the set of indices of the
intermediate wires of the circuit not being input wires of the
circuit.
During generation step E3 of a public evaluation key and a public
verification key, random variables r.sub.v, r.sub.w, s,
.alpha..sub.v, .alpha..sub.w, .alpha..sub.y, .beta., .gamma. are
first generated in .sub.q.
Then the following coefficients are defined:
r.sub.y=r.sub.vr.sub.w, g.sub.v1=g.sub.1.sup.r.sup.v,
g.sub.w1=g.sub.1.sup.r.sup.w, g.sub.w2=g.sub.2.sup.r.sup.w,
g.sub.y1=g.sub.1.sup.r.sup.y and g.sub.y2=g.sub.2.sup.r.sup.y.
The public evaluation key EK.sub.F is then generated as equal to
(EK.sub.F1, EK.sub.F2) where
.times..times..times..times..function..di-elect
cons..times..times..function..di-elect
cons..times..times..function..di-elect
cons..times..times..alpha..function..di-elect
cons..times..times..alpha..function..di-elect
cons..times..times..alpha..function..di-elect cons..di-elect
cons..delta..times..times..beta..function..times..times..beta..function..-
times..times..beta..function..di-elect cons. ##EQU00026##
.times..times..times..times..function..di-elect cons.
##EQU00026.2##
The public verification key VK.sub.F is also generated as equal to
(VK.sub.F1, VK.sub.F2) where:
VK.sub.F1=(g.sub.1,{g.sub.v1.sup.v.sup.i.sup.(s)}.sub.i.di-elect
cons.[1,.theta.],{g.sub.y1.sup.y.sup.i.sup.(s)}.sub.i.di-elect
cons.[1,.theta.])
VK.sub.F2=(g.sub.2,g.sub.2.sup..alpha..sup.v;g.sub.2.sup..alpha..sup.w,g.-
sub.2.sup..alpha..sup.y,g.sub.2.sup..beta..gamma.,g.sub.y2.sup.t(s),{g.sub-
.w2.sup.w.sup.i.sup.(s)}.sub.i.di-elect cons.[1,.theta.]).
The remote computation device then obtains E4 the arithmetic
circuit and the public evaluation key.
For each biometric datum of the entity, at least one matching value
between the biometric datum of the entity and at least one
reference biometric datum can then be determined E5 by the remote
computation device by evaluating the arithmetic circuit received
from the biometric data of the entity and the reference biometric
data. The set of values of the circuit {c.sub.i}.sub.i.di-elect
cons.[1,.rho.] can then be obtained.
Generation E6 by the remote computation device, for each determined
matching value, of a proof of correction of the computation
execution of the matching value, so-called generated proof
.pi.=(.pi..sub.1,.pi..sub.2) can then comprise:
determination of a polynomial h(x) such that p(x)=h(x)t(x) with
p(x)=(v.sub.0(x)+.SIGMA..sub.i=1.sup..rho.c.sub.iv.sub.i(x))(w.sub.0(x)+.-
SIGMA..sub.i=1.sup..rho.c.sub.iw.sub.i(x))-(x)+.SIGMA..sub.i=1.sup..rho.c.-
sub.iy.sub.i(x)),
computation of:
.pi..times..times..function..times..times..function..times..times..functi-
on..function..times..times..alpha..function..times..times..alpha..function-
..times..times..alpha..function..times..times..beta..function..times..time-
s..beta..function..times..times..beta..function. ##EQU00027## and
.pi..sub.2=(g.sub.w2.sup.W.sup.mid.sup.(s)) where:
v.sub.mid(x)=.SIGMA..sub.i.di-elect cons.I.sub.mid
c.sub.iv.sub.i(x), w.sub.mid(x)=.SIGMA..sub.i.di-elect
cons.I.sub.mid c.sub.iw.sub.i(x) and
y.sub.mid=.SIGMA..sub.i.di-elect cons.I.sub.mid c.sub.iy.sub.i(x).
The remote computation device then transmits E7 the matching values
and said generated proofs to the client device.
The proofs received by the client device are of the form
(.pi..sub.r1, .pi..sub.r12) with: .pi..sub.r1 in the form of:
(g.sub.v1.sup.V.sup.mid, g.sub.w1.sup.W.sup.mid,
g.sub.y1.sup.Y.sup.mid, g.sub.1.sup.H, g.sub.v1.sup.V'.sup.mid,
g.sub.w1.sup.W'.sup.mid, g.sub.y1.sup.Y'.sup.mid, g.sub.1.sup.Z)
and .pi..sub.r2 in the form g.sub.w2.sup.w.sup.mid.
The client device then verifies E8 each received proof
(.pi..sub.r1, .pi..sub.r2) by performing the following equality
tests:
e(g.sub.v1.sup.v.sup.o.sup.(s)g.sub.v1.sup.v.sup.io.sup.(s)g.sub.v1.sup.V-
.sup.mid,g.sub.w2.sup.w.sup.o.sup.(s)g.sub.w2.sup.w.sup.io.sup.(s)g.sub.w2-
.sup.W.sup.mid)=e(g.sub.1.sup.H,g.sub.y2.sup.t(s))e(g.sub.y1.sup.y.sup.o.s-
up.(s)g.sub.y1.sup.y.sup.io.sup.(s)g.sub.y1.sup.Y.sup.mid,g.sub.2),
e((g.sub.v1.sup.V'.sup.mid).sup.d.sup.1(g.sub.w1.sup.W'.sup.mid).sup.d.su-
p.2(g.sub.y1.sup.Y'.sup.mid).sup.d.sup.3,g.sub.2)=e((g.sub.v1.sup.V.sup.mi-
d).sup.d.sup.1,g.sub.2.sup..alpha..sup.v)e((g.sub.w1.sup.W.sup.mid).sup.d.-
sup.2,g.sub.2.sup..alpha..sup.w)e((g.sub.y1.sup.Y.sup.mid).sup.d.sup.3,g.s-
ub.2.sup..alpha..sup.y),
e((g.sub.1.sup.Z,g.sub.2.sup..gamma.)=e(g.sub.v1.sup.V.sup.midg.sub.w1.su-
p.W.sup.midg.sub.y1.sup.Y.sup.mid,g.sub.2.sup..beta..gamma.) where
g.sub.v1.sup.v.sup.io.sup.(s)=.PI..sub.i=1.sup..theta.(g.sub.v1.sup.v.sup-
.i.sup.(s)).sup.c.sup.i,
g.sub.w1.sup.w.sup.io.sup.(s)=.PI..sub.i=1.sup..theta.(g.sub.w1.sup.w.sup-
.i.sup.(s)).sup.c.sup.i, and
g.sub.y1.sup.y.sup.io.sup.(s)=.PI..sub.i=1.sup..theta.(g.sub.y1.sup.y.sup-
.i.sup.(s)).sup.c.sup.i and (d.sub.1, d.sub.2, d.sub.3) elements of
.sub.q on .lamda. bits with .lamda. a security parameter. In this
mode of operation the verification step of said received proofs
therefore comprises batch verification of pairings.
Case m>1:
Bank B is called a sub-set of indices [1, .rho.] (in other words a
sub-set of the circuit wires) and an instance of a bank B is a set
of values for these indices (for example noted
{c.sub.j}.sub.j.di-elect cons.B).
The function F is divided into .omega. sub-functions F.sub.1, . . .
, F.sub..omega.. For example in the case of FIGS. 6 and 7 selection
can be made respectively .omega.=2 and .omega.=3 as described
hereinabove.
.sigma.=((f.sub.l, (T.sub.l1, . . . , T.sub.ll))).sub.l.di-elect
cons.[1,L] is defined as a scheduling of length L with
f.sub.l.di-elect cons.{1, . . . , .omega.} the index of the
function to be computed.
By way of example, in the case m=2 or m=3 described hereinabove in
reference to FIGS. 6a to 6c, L=m+1 and the function F is split into
two functions F.sub.1, F.sub.2.
The banks used are: (B.sub.io, B.sub.L.sub.1, B.sub.L.sub.2,
B.sub.B.sub.1, . . . , B.sub.B.sub.m) where:
B.sub.io: banks of input/output type. Number of instances: m+1
B.sub.L.sub.1: bank of local type for F1. Number of instances: m
B.sub.L.sub.2: bank of local type for F2. Number of instances: 1
B.sub.B.sub.1, . . . , B.sub.B.sub.m: banks of bus type. An
instance of each. The scheduling of proofs, of length m+1, is:
.sigma.=((1,(1,1,0,1,0, . . . 0)), . . . ,(1,(m,m,0, . . .
,1)),(2,(m+1,0,1, . . . ,1))) In other words, the scheduling of
proofs is: For l.di-elect cons.{1, . . . , m}: digest of
B.sub.L.sub.1.sup.(l), pledging of B.sub.B.sub.l, proof with
B.sub.L.sub.1.sup.(l), inputs B.sub.io.sup.(l), and bus
B.sub.B.sub.l. For l=m+1: digest of B.sub.L.sub.2, proof with
B.sub.L.sub.2, inputs B.sub.io.sup.(l), and all buses
B.sub.B.sub.1, . . . , B.sub.B.sub.m. By way of example, in the
case m.gtoreq.4 described hereinabove in reference to FIGS. 7a and
7b, the function F is split into three functions F.sub.1, F.sub.2,
F.sub.3. The banks used are: (B.sub.io, B.sub.L1, B.sub.L2,
B.sub.L2, B.sub.B1, B.sub.B2) where: B.sub.io: banks of
input/output type. Number of instances: m+1 B.sub.L.sub.1: bank of
local type for F.sub.1. Number of instances: .left
brkt-top.m/2.right brkt-bot. B.sub.L.sub.2: bank of local type for
F.sub.2. Number of instances: .left brkt-top.m/2.right brkt-bot.
B.sub.L.sub.3: bank of local type for F.sub.3. Number of instances:
1 B.sub.B.sub.1, B.sub.B.sub.2: banks of bus type. .left
brkt-top.m/2.right brkt-bot. instances of the first .left
brkt-top.m/2.right brkt-bot. instances of the second. The
scheduling of proofs, of length m+1, is:
.sigma.=(.sigma..sub.1,.sigma..sub.2, . . .
.sigma..sub.m,.sigma..sub.m+1) where: For l .di-elect cons.{1, . .
. , m}: If l odd: digest of B.sub.L.sub.1.sup.(.left
brkt-top.l/2.right brkt-bot.), pledging of B.sub.B.sub.1, proof
with B.sub.L.sub.1.sup.(.left brkt-top.l/2.right brkt-bot.), inputs
B.sub.io.sup.(l), and bus B.sub.B.sub.1. .sigma..sub.l=(1,(l,.left
brkt-top.l/2.right brkt-bot.,0,0,.left brkt-top.l/2.right
brkt-bot.,.left brkt-top.l/2.right brkt-bot.-1)) If l even: digest
of B.sub.L.sub.2.sup.(.left brkt-top.l/2.right brkt-bot.), pledging
of B.sub.B.sub.2, proof with B.sub.L.sub.2.sup.(.left
brkt-top.l/2.right brkt-bot.), inputs B.sub.io.sup.(l), and bus
B.sub.B.sub.2 . . . .sigma..sub.l=(2,(l,0,.left brkt-top.l/2.right
brkt-bot.,0,.left brkt-top.l/2.right brkt-bot.-1,.left
brkt-top.l/2.right brkt-bot.)) For l=m+1: digest of B.sub.L.sub.3,
proof with B.sub.L.sub.3, inputs B.sub.io.sup.(l), and the bus bank
B.sub.B.sub.1 (if m is odd) or B.sub.B.sub.2 (if not).
.sigma..sub.l=(3,(m+1,0,0,1,.left brkt-top.l-1/2.right
brkt-bot.,0)) or .sigma..sub.l=(3,+1,0,0,1,0,.left
brkt-top.l-1/2.right brkt-bot.)). For a number x the notation .left
brkt-top.x/2.right brkt-bot. (respectively .left brkt-bot.x/2.right
brkt-bot.) designates the natural integer greater than or equal
(respectively less then or equal) to the rational value x/2. For
more information on the use of banks and such scheduling, the
paragraphs hereinbelow can be viewed in the light of the
publication referenced hereinabove describing the Geppetto protocol
from which the protocol presented hereinbelow is derived.
In these second and third modes of operation, an asymmetric
bilinear environment (q, G.sub.1, G.sub.2, G.sub.T, g.sub.1,
g.sub.2, e) is defined with q a prime number, G.sub.1, G.sub.2 and
G.sub.T three groups of order q, g.sub.1 a generator of G.sub.1,
g.sub.2 a generator of G.sub.2, and e a non-degenerate bilinear
pairing e: G.sub.1.times.G.sub.2.fwdarw.G.sub.T.
The arithmetic circuit can be represented in the form of a
multi-QAP Q=({B.sub.b}.sub.b.di-elect cons.[1,l], t, V, W, Y) of
size .rho. and degree .delta., with {B.sub.b}.sub.b.di-elect
cons.[1,l] a set of l banks B.sub.b of Q used in computing the
function F, and V={vi}, W={wi}, Y={yi} with
0.ltoreq.i.ltoreq..rho..
During the generation step E3 by the client device of a public
evaluation key and a public verification key, random variables
s,{(.alpha..sub.bv, .alpha..sub.bw, .alpha..sub.by, .beta..sub.b,
.gamma..sub.b)}.sub.b.di-elect cons.[1,l], r.sub.v, r.sub.w are
generated in .sub.q.
Next, the following coefficients are defined:
r.sub.y=r.sub.vr.sub.w, g.sub.v1=g.sub.1.sup.r.sup.v,
g.sub.v2=g.sub.2.sup.r.sup.v, g.sub.w1=g.sub.1.sup.r.sup.w,
g.sub.w2=g.sub.2.sup.r.sup.w, g.sub.y1=g.sub.1.sup.r.sup.y and
g.sub.y2=g.sub.2.sup.r.sup.y.
The public evaluation key EK.sub.F is generated as equal to:
({EK.sub.Fb}.sub.b.di-elect
cons.[1,l],{g.sub.1.sup.s.sup.i}.sub.i.di-elect
cons.[1,.delta.],g.sub.v1.sup.t(s),g.sub.w1.sup.t(s),g.sub.y1.sup.t(s),g.-
sub.v2.sup.t(s),g.sub.w2.sup.t(s),g.sub.y2.sup.t(s)) Each public
bank key EK.sub.Fb is equal to (EK.sub.Fb1, EK.sub.Fb2) and
generated by computing:
.times..times..times..times..function..times..times..function..times..tim-
es..function..times..times..alpha..function..times..times..alpha..function-
..times..times..alpha..function..times..times..beta..function..times..time-
s..beta..function..times..times..beta..function..di-elect
cons..times..times..alpha..function..times..times..alpha..function..times-
..times..alpha..function..times..times..beta..function..times..times..beta-
..function..times..times..beta..function. ##EQU00028##
.times..times..times..times..times..function..di-elect
cons..times..times..alpha..function. ##EQU00028.2##
The public verification key VK.sub.F is also generated as equal to:
({VK.sub.Fb}.sub.b.di-elect cons.[1,l], g.sub.1, g.sub.2,
g.sub.y2.sup.t(s)). Each public bank key VK.sub.Fb is equal to
(g.sub.2.sup..alpha..sup.bv, g.sub.2.sup..alpha..sup.bw,
g.sub.2.sup..alpha..sup.by, g.sub.2.sup..gamma..sup.b,
g.sub.2.sup..beta..sup.b.sup..gamma..sup.b).
The remote computation device obtains E4 the arithmetic circuit and
the public evaluation key.
For each biometric datum of the entity, at least one matching value
between said biometric datum and at least one reference biometric
datum can then be determined E5 by the remote computation device by
evaluating the arithmetic circuit received from the biometric data
of the entity and the reference biometric data. The remote
computation device evaluates each sub-function F.sub..omega. from
the biometric data of the entity and the reference biometric data
for obtaining the matching value and the values of the circuit.
Generation E6 by the remote computation device, for each determined
matching value, of a proof of correction of the computation
execution of the matching value can comprise for each l={1, . . . ,
L} a list of digests and proofs obtained as described
hereinbelow.
Let .LAMBDA.[1,l] be the set of indices b.di-elect cons.[1,l] such
that T.sub.lb.noteq.0 in the scheduling .sigma.=((f.sub.l,
(T.sub.l1, . . . , T.sub.ll))).sub.l.di-elect cons.[1,L].
Hereinbelow the following: .GAMMA.=U.sub.b.SIGMA..LAMBDA.B.sub.b,
{c.sub.j}.sub.j.di-elect cons.B.sub.b an instance of the bank
B.sub.b and .DELTA.={c.sub.i}.sub.i.di-elect cons..GAMMA. the set
of values of .GAMMA..
For each bank B.sub.b such as b.di-elect cons..LAMBDA.,
the remote computation device generates pledging random variables
o.sub.b=(o.sub.bv, o.sub.bw, o.sub.by) in .sub.q.
it then computes the digest D.sub.b equal to (D.sub.b1, D.sub.b2)
from the instance of the bank of variables B.sub.b:
B.sub.b.sup.(T.sup.lb.sup.)={c.sub.i.di-elect
cons..DELTA.}.sub.i.di-elect cons.B.sub.b and pledging random
variables o.sub.b. Such digests are such that: if the bank B.sub.b
is an input/output bank: D.sub.b1=(g.sub.v1.sup.v.sup.(b).sup.(s),
g.sub.y1.sup.y.sup.(b).sup.(s)) and
D.sub.b2=(g.sub.w1.sup.w.sup.(b).sup.(s)), if the bank B.sub.b is
not an input/output bank:
.times..times..times..times..function..times..times..function..times..tim-
es..function..times..times..alpha..function..times..times..alpha..function-
..times..times..alpha..function..times..times..beta..function..times..time-
s..beta..function..times..times..beta..function. ##EQU00029##
.times..times..times..times..function. ##EQU00029.2## with:
v.sup.(b)(s)=.SIGMA..sub.i.di-elect
cons.B.sub.bc.sub.iv.sub.i(s)+o.sub.bvt(s),
w.sup.(b)(s)=.SIGMA..sub.i.di-elect
cons.B.sub.bc.sub.iw.sub.i(s)+o.sub.bwt(s),
y.sup.(b)(s)=.SIGMA..sub.i.di-elect
cons.B.sub.bc.sub.iy.sub.i(s)+o.sub.byt(s), The remote computation
device then determines a polynomial h.sup.(l)(x) such that
p.sup.(l)(x)=h.sup.(l)(x)t(x) with
p.sup.(l)(x)=(v.sub.0(x)+.SIGMA..sub.i.di-elect
cons..GAMMA.c.sub.iv.sub.j(x)+.SIGMA..sub.b.di-elect
cons..LAMBDA.o.sub.bvt(x))(w.sub.0(x)+.SIGMA..sub.i.di-elect
cons..GAMMA.c.sub.iw.sub.j(x)+.SIGMA..sub.b.di-elect
cons..LAMBDA.o.sub.bwt(x))-(y.sub.0(x)+.SIGMA..sub.i.di-elect
cons..GAMMA.c.sub.iy.sub.j(x)+.SIGMA..sub.b.di-elect
cons..LAMBDA.o.sub.byt(x)) Finally, it computes a proof element
.pi..sup.(l) equal to g.sub.1.sup.h.sup.(l).sup.(s).
The remote computation device then transmits E7 the matching values
and said generated proofs comprising the list of computed digests
and proof elements to the client device.
The proofs received by the client device are of the form of:
D.sub.1.sup.(1), . . . , D.sub.l.sup.(1), .pi..sup.(1), . . . ,
D.sub.1.sup.(L), . . . , D.sub.l.sup.(L), .pi..sup.(L) where for
all l.di-elect cons.{1, . . . , L} and .di-elect cons.{1, . . .
,l}:
.times..times..times..times..times..times..times..times.'.times..times..t-
imes..times.'.times..times..times..times.'.times..times..times..times.
##EQU00030## and .pi..sup.(l)=g.sub.1.sup.H.sup.(l).
Two verification implementation variants of the received proof E8
are specified hereinbelow.
In a first implementation variant, the client device then verifies
each received proof by performing: verification of Le digests, for
l.di-elect cons.{1, . . . , L} and b.di-elect cons.{1, . . . , l}
comprising the following equality tests:
.function..times..times.'.times..times..alpha..function..times..times.
##EQU00031##
.function..times..times.'.function..alpha..function..times..times.
##EQU00031.2##
.function..times..times.'.times..times..alpha..function..times..times.
##EQU00031.3##
.function..gamma..function..times..times..times..times..times..times..bet-
a..gamma. ##EQU00031.4## verification of L proofs comprising for
l.di-elect cons.{1, . . . , L} the following equality test:
.function. .times..times..times.
.times..times..times..times..function..times..times..function..function.
.times..times..times. ##EQU00032##
In a second implementation variant, the client device then verifies
each received proof by executing batch verification comprising,
given a correction parameter .lamda.: selection of a random vector
(d.sub.1, . . . , d.sub.3l) of elements of size .lamda., batch
verification of the Ll digests, in l times by executing the
following equality tests, for b.di-elect cons.{1, . . . , l}:
.function..times..times..times..times.'.function..alpha..function..times.-
.times..times.'.function..alpha..function..times..times..times..times.'.fu-
nction..alpha..function..times..times..times..times..times..times..times..-
times..times..times..times..times. ##EQU00033##
.times..function..times..times..gamma..function..times..times..times..tim-
es..times..times..times..times..beta..gamma. ##EQU00033.2## batch
verification the L proofs by executing the following equality
test:
.times..function. .times..times..times..times.
.times..times..times..function..times..times..times..function..function..-
times. .times..times..times. ##EQU00034## and as a verification
option of the belonging of elements on which the pairings is
applied to their respective groups.
The method performed carries out biometric identification by
comparing biometric data in terms of the scope of a publicly
verifiable computation protocol and minimizing the time necessary
for production and verification of proofs relative to proper
execution of this computation, by way of minimization of the number
of multipliers employed to represent this computation in the form
of an arithmetic circuit.
* * * * *